problems solving exercises in physics: conceptual...

Exercises in Physics

Jennifer Bond Hickman

Needham, MassachusettsUpper Saddle River, New Jersey

Glenview, Illinois

To my grandfather, C. Lawrence BondWhen I was 10 years old, you paid me 10¢ to write a book for you. I’ve finally finished it!

Illustrations by Jennifer Bond Hickman.

Cover Photograph: Motor Press Agent/Superstock, Inc.

Many of the designations used by manufacturers and sellers to distinguishtheir products are claimed as trademarks. Where such a designation appearsin this book, and the publisher was aware of a trademark claim, thedesignations have been printed in initial caps (e.g., Macintosh).

Copyright © 2002 by Prentice-Hall, Inc., Upper Saddle River, New Jersey07458. All rights reserved. Printed in the United States of America. Thispublication is protected by copyright, and permission should be obtainedfrom the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical,photocopying, recording, or likewise. For information regardingpermission(s), write to: Rights and Permissions Department.

ISBN 0-13-054275-X

26 V031 13 12 11

Contents

iii

Preface to Students:Welcome to Physics! iv

1 Motion 11-1 Speed, Velocity, and

Acceleration 11-2 Free Fall 8

2 Vectors and Projectiles 152-1 Vectors and Scalars 152-2 Projectile Motion 21

3 Forces 293-1 Forces and Acceleration 293-2 Friction 353-3 Statics 383-4 Pressure 44

4 Momentum 514-1 Impulse and Momentum 514-2 Conservation of Momentum 55

5 Energy and Machines 635-1 Work and Power 635-2 Energy 665-3 Machines and Efficiency 72

6 Circular and Rotational Motion 816-1 Centripetal Acceleration and

Force 816-2 Torque 876-3 Moment of Inertia and

Angular Momentum 91

7 Law of Universal Gravitation 977-1 Gravitational Force 977-2 Gravitational Acceleration 1017-3 Escape Speed 104

8 Special Relativity 1098-1 Time Dilation 1098-2 Relativistic Length and

Energy 113

9 Solids, Liquids, and Gases 1199-1 Density 1199-2 Solids 1219-3 Liquids 1249-4 Gases 130

10 Temperature and Heat 13510-1 Temperature and Expansion 13510-2 Heat 140

11 Simple Harmonic Motion 14911-1 Springs 14911-2 Pendulums 153

12 Waves and Sound 15912-1 Wave Motion 15912-2 Doppler Effect 16112-3 Standing Waves 165

13 Reflection and Refraction 17113-1 The Speed of Light 17113-2 Reflection 17313-3 Refraction 177

14 Lenses, Diffraction, and Interference 18314-1 Lenses, Telescopes, and

Magnifying Glasses 18314-2 Eyeglasses 18914-3 Diffraction and Interference 192

15 Electrostatics 19715-1 Electrostatic Force 19715-2 Electric Field 20015-3 Electrical Potential

Difference 203

16 Direct Current Circuits 20916-1 Current and Resistance 20916-2 Capacitance 21216-3 Power 21416-4 Series and Parallel Circuits 217

17 Magnetism and ElectromagneticInduction 22517-1 Magnetic Forces and Fields 22517-2 Electromagnetic Induction 227

18 Modern Physics 23318-1 The Atom and the Quantum 23318-2 The Photoelectric Effect 23618-3 Energy Level Diagrams 23918-4 Radioactivity 241

Appendix A: Working With Numbers 247Significant FiguresUnit ConversionsSome Simple Trigonometry

RelationshipsSome Common Prefixes

Appendix B: Selected Answers 249

iv

Welcome to Physics!Studying physics is exciting because it can help you answer many questionsabout how and why our world works. Your workbook is designed to takesome “real-life” situations and examine them with the use of equations, a taskoften referred to as problem solving. Problem solving, however, is more thanjust solving numerical exercises by doing calculations. Using mathematics isonly one way to obtain a solution. Another effective method of problemsolving involves drawing on conceptual understanding to explain how theworld works and applying those concepts in the laboratory. Like scientists, weperform experiments to test our hypotheses. Until we can understand theconcepts and have the opportunity to make our own discoveries, the numbersand equations of physics are meaningless. In the words of Paul G. Hewitt,author of Conceptual Physics, “Formulas [should be used] as guides tothinking. . . . We [must] learn to conceptualize before we learn to compute.”

This book is not meant to stand alone. It is not meant to replace your physicstext, the laboratory work that you do, or your physics teacher. Its purpose isto reinforce the concepts that you have already learned in class and to giveyou the opportunity to try some calculations with your teacher’s help. If youhave had difficulty solving word problems in the past, rely on yourconceptual understanding of the physics to reason out what should behappening before beginning your mathematical solution. The procedureoutlined in the next section will lead you step-by-step through the exercisesand make learning to do simple computations a little easier.

How to Use This Book

As you begin to use this book, you will discover that the word problem hasbeen replaced with the word exercise. A physics exercise does not reallybecome a problem until you accept the challenge it offers and attempt to solveit. Once you have chosen to make it your problem, you have a personalinterest in finding the solution.

Each chapter of this workbook is divided into two or more topic sections thatbegin with some physics theory. This theory section provides a very briefreview of the concepts and equations your teacher has discussed in class, andis not an introduction to new material. It is presumed that you have alreadylearned everything in the theory section before beginning the exercises. Thisreview is simply a reminder and a place to find all the equations you need.

Following the theory, there is a section called Solved Examples, where thetheory is applied to exercises similar to those you will be expected to solvelater. Solutions are organized to make it easy to follow a calculation frombeginning to end. Most solved examples are in the following format.

Given: States the known values Unknown: Lists the unknown you are in the exercise. looking for.

Original equation: Shows the equationin its original form.

Solve: Shows the equation set up in terms of the unknown, substitutes thenumerical values, and solves for the unknown. The answer is thenwritten with the correct units and shown in boldface type for easyidentification.

A section of Practice Exercises allows you to apply some of the skills you havelearned to new situations.

For more practice, at the end of each chapter there is a section of AdditionalExercises, which require the same level of understanding as the PracticeExercises. The final section, called Challenge Exercises for Further Study,contains exercises requiring more complex calculations. Challenge Exercisesare intended for you to use after you have mastered the skills used in earlierexercises and are anxious to take on some more rigorous computations.

At the end of the workbook, some Selected Answers will allow you to checkyour progress.

Using the Right RecipeSolving physics exercises is much like baking a cake. The first time you try todo it, you must read the recipe very carefully and use exactly the ingredientslisted. The next time, you are a little less nervous about how well the cake willturn out. Pretty soon you can make the cake without having to read the recipeat all. You eventually become so comfortable making cakes that you are ableto experiment by adding ingredients in a different order or changing therecipe slightly to make the cake even better. When solving physics exercises,you will find it easy to follow the prescribed “recipe” shown in the SolvedExamples. After trying a few exercises, you will have started to develop astrategy for constructing your solution that you can retain throughout theentire book. As you get better and better at doing calculations and youdevelop a greater conceptual understanding of the physics involved, you mayeven come up with an alternative method of solving an exercise that isdifferent from the one used in this book. If so, congratulations! You have donejust what the physicist does when he or she tries to find a solution. Be sure toshow your teacher and classmates your alternative approach. It is valuable tolook at many different solutions to the same exercise.

An Alternative to Counting on Your FingersEarly scientists had to make all of their calculations by hand. Later, the sliderule made calculations a little quicker. Today’s tool is the hand-held pocketcalculator. To save time, you are encouraged to do your calculations with theuse of a calculator, but be sure that you first understand why you are doingthem. Remember, it’s important to know how to operate without a calculatoras well. Many students rely so heavily on their calculator that they begin tolose the skill of doing calculations by hand. It is extremely important to beable to add, subtract, multiply, divide, and square numbers. You should

v

practice working with exponents (called scientific notation) and estimatinganswers to the nearest power of ten because you may not always have acalculator handy!

How Much is Too Much?When making measurements, you may have measurement tools that allowyou only a certain degree of precision. For example, you may be able tomeasure your friend’s height to the nearest millimeter, but estimating it anycloser is difficult. You may say his height is 1536 mm or, in other words, 1.536 m. Since we don’t know what comes after the 6, we say that this numbercontains 4 significant figures. Each one can be accurately measured. Whenadding, subtracting, multiplying, and dividing numbers, it is important tokeep significant figures in mind. The invention of the calculator has made thistask difficult, because the calculator customarily carries out our calculations to8 figures or more, many of which are probably not significant. The rules forthe correct use of significant figures can be found in Appendix A. You willfind that all of the solved examples in this book and the selected answers inthe back adhere to these rules on significant figures and you should too,whenever possible.

You Can’t Add Apples and OrangesWhen solving numerical exercises, it is always important to include theproper units with any number you are using. Not only will this help youdetermine the units in the final answer, but it will also help you with yournumerical solution as well. If the units in an exercise do not combine to givethe correct units in your final answer, then you may have made a mistake insetting up the original equation. In other words, using the correct units is away of double-checking all of your work.

In this book we will use the units of the Système International (SI), the standardsystem of units in the physics community. Any units not written in the SIform should be converted to the SI system before beginning your calculations.See Appendix A for a review of some important prefixes that you will seewhen working in the SI system.

A Word of ThanksI would like to thank the physics students at BostonUniversity Academy, Phillips Academy, and BelmontHigh School for their input in writing, editing andsolving exercises in this book.

Finally, I give my heartfelt thanks to my husband, PaulHickman, for his countless hours proofreading, editing,and problem solving, and for his unending support andencouragement throughout my work on this book.

vi

1 Motion

1

1-1 Speed, Velocity, and Acceleration

Speed vs. Velocity

Vocabulary Distance: How far something travels.

Vocabulary Displacement: How far something travels in a given direction.

Notice that these two terms are very similar. Distance is an example of whatwe call a scalar quantity. In other words, it has magnitude, but no direction.Displacement is an example of a vector quantity because it has bothmagnitude and direction.

The SI (Système International) unit for distance and displacement is the meter (m).

Displacements smaller than a meter may be expressed in units of centimeters(cm) or millimeters (mm). Displacements much larger than a meter may beexpressed in units of kilometers (km). See Appendix A for the meanings ofthese and other common prefixes.

Vocabulary Speed: How fast something is moving.

average speed � or vav �

Vocabulary Velocity: How fast something is moving in a given direction.

average velocity � or vav � �

where df and tf are the final position and time respectively, and do and to arethe initial position and time. The symbol “�” (delta) means “change” so �d isthe change in position, or the displacement, while �t is the change in time.

In this book all vector quantities will be introduced in an equation with boldtype while all scalar quantities will be introduced in an equation in regulartype. Note that speed is a scalar quantity while velocity is a vector quantity.

df � do

tf � to

�d�t

displacement

elapsed time

d�t

distance traveledelapsed time

The SI unit for both speed and velocity is the meter per second (m/s).

When traveling in any moving vehicle, you rarely maintain the same velocitythroughout an entire trip. If you did, you would travel at a constant speed ina straight line. Instead, speed and direction usually vary during your time oftravel.

If you begin and end at the same location but you travel for a great distance ingetting there (for example, when you travel in a circle), you have a measurableaverage speed. However, since your total displacement for such a trip is zero,your average velocity is also zero. In this chapter, both average speed andaverage velocity will be written as vav. The “av” subscript will be dropped inlater chapters.

Acceleration

Vocabulary Acceleration: The rate at which the velocity changes during a given amountof time.

acceleration � or a � �

where the terms vf and vo mean final velocity and initial velocity, respectively.

The SI unit for acceleration is the meter per second squared (m/s2).

If the final velocity of a moving object is smaller than its initial velocity, theobject must be slowing down. A slowing object is sometimes said to havenegative acceleration because the magnitude of the acceleration is preceded by anegative sign.

Solved Examples

Example 1: Benjamin watches a thunderstorm from his apartment window. He sees theflash of a lightning bolt and begins counting the seconds until he hears the clapof thunder 10. s later. Assume that the speed of sound in air is 340 m/s. Howfar away was the lightning bolt a) in m? b) in km? (Note: The speed of light,3.0 � 108 m/s, is considerably faster than the speed of sound. That is why yousee the lightning flash so much earlier than you hear the clap of thunder. Inactuality, the lightning and thunder clap occur almost simultaneously.)

a. Given: vav � 340 m/s Unknown: �d � ?�t � 10.0 s Original equation: vav �

Solve: �d � vav�t � (340 m/s)(10. s) � 3400 m

�d�t

vf � vo

tf � to

�v�t

change in velocity

elapsed time

2 Motion

b. For numbers this large you may wish to express the final answer in kmrather than in m. Because “kilo” means 1000, then 1.000 km � 1000. m.

3400 m � 3.4 km

The lightning bolt is 3.4 km away, which is just a little over two miles forthose of you who think in English units!

Example 2: On May 28, 2000, Juan Montoya became the first Colombian citizen to win theIndianapolis 500. Montoya completed the race in a time of 2.98 h. What wasMontoya’s average speed during the 500.-mi race? (Note: Generally the unit“miles” is not used in physics exercises. However, the Indianapolis 500 is arace that is measured in miles, so the mile is appropriate here. Don’t forget,the SI unit for distance is the meter.)

Given: d � 500. mi Unknown: vav � ?�t � 2.98 h Original equation: �t �

Solve: �t � � � 168 mi/h

Example 3: The slowest animal ever discovered was a crab found in the Red Sea. It traveled with an average speed of 5.70 km/y.How long would it take this crab to travel 100. km?

Given: �d � 100. km Unknown: �t � ?vav � 5.70 km/y Original equation: �t �

Solve: �t � � � 17.5 y A long time!

Example 4: Tiffany, who is opening in a new Broadway show, has some limo trouble inthe city. With only 8.0 minutes until curtain time, she hails a cab and theyspeed off to the theater down a 1000.-m-long one-way street at a speed of 25 m/s. At the end of the street the cab driver waits at a traffic light for 1.5 min and then turns north onto a 1700.-m.-long traffic-filled avenue onwhich he is able to travel at a speed of only 10.0 m/s. Finally, this brings themto the theater. a) Does Tiffany arrive before the theater lights dim? b) Draw adistance vs. time graph of the situation.

Solution: First, break this exercise down into segments and solve eachsegment independently.

Segment 1: (one-way street)

Given: �d � 1000. m Unknown: �t � ?vav � 25 m/s Original equation: vav �

Solve: �t � � � 40. s1000. m25 m>s

�dvav

�d�t

100. km5.70 km>y

�dvav

�dvav

500. mi2.98 h

�dv

�dv

11.000 km 21000. m

Motion 3

Segment 2: (traffic light)

Given: �t � 1.5 min (1.5 min) � 90. s

Segment 3: (traffic-filled avenue)

Given: �d � 1700. m Unknown: �t � ?vav � 10.0 m/s Original equation: vav �

Solve: �t � � � 170. s

total time � 40. s � 90. s � 170. s � 300. s (300. s) � 5.0 min

Yes, she not only makes it to the show in time, but she even has 3.0 minutes tospare to put on her costume and make-up.

b. The motion of the cab can be described by the following graph.

In Segment 1, the distance of 1000. m was covered in a fairly short amount oftime, which means that the cab was traveling quickly. This high speed can beseen as a steep slope on the graph.

In Segment 2, the cab was at rest. Notice that even though the cab did notmove, time continued on, resulting in a horizontal line on the graph.

In Segment 3, the distance of 1700. m was covered in a much longer amountof time so the cab was traveling slowly. This low speed is indicated by a slopethat is not as steep as that in segment 1.

Remember, all graphs should have titles and the axes should be labeled withthe correct units.

11.0 min 2160. s 2

1700. m10.0 m>s

�dvav

�d�t

160. s 211.0 min 2

4 Motion

Example 5: Grace is driving her sports car at 30 m/s when a ball rolls out into the streetin front of her. Grace slams on the brakes and comes to a stop in 3.0 s. Whatwas the acceleration of Grace’s car?

Given: vo � 30 m/s Unknown: a � ?vf � 0 m/s Original equation: a ��t � 3.0 s

Solve: a � � � �10 m/s2

The negative sign means the car was slowing down.

Practice Exercises

Exercise 1: Hans stands at the rim of the Grand Canyon and yodels down to the bottom.He hears his yodel echo back from the canyon floor 5.20 s later. Assume thatthe speed of sound in air is 340.0 m/s. How deep is the canyon at thislocation?

Answer:

Exercise 2: The world speed record on water was set on October 8, 1978 by Ken Warby ofBlowering Dam, Australia. If Ken drove his motorboat a distance of 1000. m in7.045 s, how fast was his boat moving a) in m/s? b) in mi/h?

Answer: a.

Answer: b.

0 m>s � 30 m>s3.0 s

vf � vo

�t

vf � vo

�t

Motion 5

Exercise 3: According to the World Flying Disk Federation, on April 8, 2000, JenniferGriffin of Fredericksburg, Virginia threw a Frisbee for a distance of 138.56 mto capture the women’s record. If the Frisbee was thrown horizontally with aspeed of 13.0 m/s, how long did the Frisbee remain aloft?

Answer:

Exercise 4: It is now 10:29 a.m., but when the bell rings at 10:30 a.m. Suzette will be latefor French class for the third time this week. She must get from one side of theschool to the other by hurrying down three different hallways. She runs downthe first hallway, a distance of 35.0 m, at a speed of 3.50 m/s. The secondhallway is filled with students, and she covers its 48.0-m length at an averagespeed of 1.20 m/s. The final hallway is empty, and Suzette sprints its 60.0-mlength at a speed of 5.00 m/s. a) Does Suzette make it to class on time or doesshe get detention for being late again? b) Draw a distance vs. time graph ofthe situation.

Answer: a.

Exercise 5: A jumbo jet taxiing down the runway receives word that it must return to thegate to pick up an important passenger who was late to his connecting flight.The jet is traveling at 45.0 m/s when the pilot receives the message. What is theacceleration of the plane if it takes the pilot 5.00 s to bring the plane to a halt?

Answer:

6 Motion

Exercise 6: While driving his sports car at 20.0 m/s down a four-lane highway, Eddiecomes up behind a slow-moving dump truck and decides to pass it in the left-hand lane. If Eddie can accelerate at 5.00 m/s2, how long will it take for himto reach a speed of 30.0 m/s?

Answer:

Exercise 7: Vivian is walking to the hairdresser’s at 1.3 m/s when she glances at her watchand realizes that she is going to be late for her appointment. Vivian graduallyquickens her pace at a rate of 0.090 m/s2. a) What is Vivian’s speed after 10.0 s? b) At this speed, is Vivian walking, jogging, or running very fast?

Answer: a.

Answer: b.

Exercise 8: A torpedo fired from a submerged submarine is propelled through the waterwith a speed of 20.00 m/s and explodes upon impact with a target 2000.0 maway. If the sound of the impact is heard 101.4 s after the torpedo was fired,what is the speed of sound in water? (Because the torpedo is held at a constantspeed by its propeller, the effect of water resistance can be neglected.)

Answer:

Motion 7

1-2 Free FallVocabulary Free Fall: The movement of an object in response to a gravitational attraction.

When an object is released, it falls toward the earth due to the gravitationalattraction the earth provides. As the object falls, it will accelerate at a constantrate of 9.8 m/s2 regardless of its mass. However, to make calculations moreexpedient and easier to do without a calculator, this number will be written asg � 10.0 m/s2 throughout this book.

There are many different ways to solve free fall exercises. The sign conventionused may be chosen by you or your teacher. In this book, the downwarddirection will be positive, and anything falling downward will be written witha positive velocity and position; anything moving upward will be representedwith a negative velocity and position. Remember: Gravity always acts to pullan object down, so the gravitational acceleration, g, will always be written as apositive number regardless of which direction the object is moving.

The displacement of a falling object in a given amount of time is written as

�d � vo�t � g�t2

The final velocity of a falling object can be represented by the equation

vf2 � vo

2 � 2g�d

or by the earlier equation, a � (vf � vo)/�t, rewritten as vf � vo � a�t, or

vf � vo � g�t

Note that the term “g” in all of these exercises can be written as “a” if you usea constant acceleration other than gravity. Therefore, these equations can beused for objects moving horizontally as well as vertically.

It is common to neglect air resistance in most free fall exercises (includingthose in this book), although in real life, air resistance is a factor that must betaken into account. This book will also assume that the initial speed of allobjects in free fall is zero, unless otherwise specified.

Solved Examples

Example 6: King Kong carries Fay Wray up the 321-m-tall Empire State Building. At thetop of the skyscraper, Fay Wray’s shoe falls from her foot. How fast will theshoe be moving when it hits the ground?

a 12b

8 Motion

Given: vo � 0 m/s Unknown: vf � ?g � 10.0 m/s2 Original equation: vf

2 � vo2 � 2g�d

�d � 321 m

Solve: vf � � �

� 80.1 m/s

Example 7: The Steamboat Geyser in Yellowstone National Park, Wyoming is capable ofshooting its hot water up from the ground with a speed of 48.0 m/s. Howhigh can this geyser shoot?

Solution: Remember, the geyser is shooting up; therefore it must have anegative initial velocity.

Given: vo � �48.0 m/s Unknown: �d � ?vf � 0 m/s Original equation: vf

2 � vo2 � 2g�d

g � 10.0 m/s2

Solve: �d � � � �115 m

As you might expect, the final answer has a negative displacement. Thismeans that the total distance the water has traveled is measured up from theground.

Example 8: A baby blue jay sits in a tall tree awaiting the arrival of its dinner. As the mother lands on the nest, she drops a worm toward the hungry chick’s mouth, but theworm misses and falls from the nest to theground in 1.50 s. How high up is the nest?

Given: vo � 0 m/s Unknown: �d � ?g � 10.0 m/s2 Original equation: �d � vo�t � g�t2

t � 1.50 s

Solve: �d � vo�t � g�t2 � 0 � (10.0 m/s2)(1.50 s)2 � 11.3 m

Example 9: A giraffe, who stands 6.00 m tall, bites a branch off a tree to chew on theleaves, and he lets the branch fall to the ground. How long does it take thebranch to hit the ground?

Given: �d � 6.00 m Unknown: �t � ?g � 10.0 m/s2 Original equation: �d � vo�t � g�t2

vo � 0 m/s

Solve: �t � � � � 1.10 s21.20 s2B216.00 m 210.0 m>s2B2¢d

g

a12b

a 12ba 1

2b

a12b

10 m>s 22 � 1�48.0 m>s 222110.0 m>s2 2

vf2 � vo

2

2g

26420 m2>s220 � 2110.0 m>s2 2 1321 m 22vo2 � 2g�d

Motion 9

Practice Exercises

Exercise 9: Billy, a mountain goat, is rock climbing on his favorite slope one sunny springmorning when a rock comes hurtling toward him from a ledge 40.0 m above.Billy ducks and avoids injury. a) How fast is the rock traveling when it passesBilly? b) How does this speed compare to that of a car traveling down thehighway at the speed limit of 25 m/s (equivalent to 55 mi/h)?

Answer: a.

Answer: b.

Exercise 10: Reverend Northwick climbs to the church belfry one morning to determinethe height of the church. From an outside balcony he drops a book andobserves that it takes 2.00 s to strike the ground below. a) How high is thebalcony of the church belfry? b) Why would it be difficult to determine theheight of the belfry balcony if the Reverend dropped only one page out of thebook?

Answer: a.

Answer: b.

Exercise 11: How long is Tina, a ballerina, in the air when she leaps straight up with aspeed of 1.8 m/s?

Answer:

10 Motion

2800. m2>s2

Exercise 12: In order to open the clam it catches, a seagull will drop the clam repeatedlyonto a hard surface from high in the air until the shell cracks. If a seagull fliesto a height of 25 m, how long will the clam take to fall?

Answer:

Exercise 13: At Six Flags Great Adventure Amusement Park in New Jersey, a popular rideknown as “Free Fall” carries passengers up to a height of 33.5 m and dropsthem to the ground inside a small cage. How fast are the passengers going atthe bottom of this exhilarating journey?

Answer:

Exercise 14: A unique type of basketball is played on the planet Zarth. During the game, aplayer flies above the basket and drops the ball in from a height of 10 m. If theball takes 5.0 s to fall, find the acceleration due to gravity on Zarth.

Answer:

Motion 11

22125 m 2 > 110.0 m>s2 222¢d>g

2670. m2>s2

Additional Exercises

A-1: During an Apollo moon landing, reflecting panels were placed on the moon.This allowed earth-based astronomers to shoot laser beams at the moon’ssurface to determine its distance. The reflected laser beam was observed 2.52 safter the laser pulse was sent. If the speed of light is 3.00 � 108 m/s, what wasthe distance between the astronomers and the moon?

A-2: The peregrine falcon is the world’s fastest known bird and has been clockeddiving downward toward its prey at constant vertical velocity of 97.2 m/s. Ifthe falcon dives straight down from a height of 100. m, how much time doesthis give a rabbit below to consider his next move as the falcon begins hisdescent?

A-3: The Kentucky Derby, the first of three horse races for the triple crown, waswon on May 7, 2000 by Fusaichi Pegasus with a time of 121.1 s. If the racecovers 2011.25 m, what was Fusaichi Pegasus’ average speed in a) m/s? b) mi/h?

A-4: For years, the posted highway speed limit was 88.5 km/h (55 mi/h) but nowsome rural stretches of highway have posted speed limits of 104.6 km/h (65 mi/h). In Maine, the distance from Portland to Bangor is 215 km. Howmuch time can be saved in traveling from Portland to Bangor at this higherspeed limit?

A-5: A tortoise and a hare are in a road race to defend the honor of their breeds.The tortoise crawls the entire 1000.-m distance at a speed of 0.2000 m/s whilethe rabbit runs the first 200.0 m at 2.000 m/s. The rabbit then stops to take anap for 1.300 h and awakens to finish the last 800.0 m with an average speedof 3.000 m/s. a) Who wins the race and by how much time? b) Draw a graphof distance vs. time for the situation.

A-6: Two physics professors challenge each other to a 100.-m race across thefootball field. The loser will grade the winner’s physics labs for one month.Dr. Nelson runs the race in 10.40 s. Dr. Bray runs the first 25.0 m with anaverage speed of 10.0 m/s, the next 50.0 m with an average speed of 9.50 m/s,and the last 25.0 m with an average speed of 11.1 m/s. Who gets stuckgrading physics labs for the next month?

A-7: A caterpillar crawling up a leaf slows from 0.75 cm/s to 0.50 cm/s at a rate of�0.05 cm/s2. How long does it take the caterpillar to make the change?

A-8: In the Wizard of Oz, Dorothy awakens in Munchkinland where her house hasbeen blown by a tornado. If the house fell from a height of 3000. m, with whatspeed did it hit the Wicked Witch of the East when it landed?

A-9: The Tambora volcano on the island of Sumbawa, Indonesia has been knownto throw ash into the air with a speed of 625 m/s during an eruption. a) Howhigh could this volcanic plume have risen? b) On Venus, where theacceleration due to gravity is slightly less than on Earth, would this volcanicplume rise higher or not as high as it does on Earth?

12 Motion

A-10: Chief Boolie, the jungle dweller, is out hunting for dinner when a coconut fallsfrom a tree and lands on his toe. If the nut fell for 1.4 s, how fast was ittraveling when it hit Chief Boolie’s toe?

A-11: Here is a bet that you are almost sure to win! Try dropping a dollar bill through a friend’s fingers and offer to let her keep it ifshe can catch it. The bill should be started just at the finger leveland your friend shouldn’t have any advanced warning when it isgoing to drop. A dollar bill has a length of 15.5 cm and humanreaction time is rarely less than 0.20 s. Do the necessarycalculations—why is this almost a sure bet?

A-12: While repairing a defective radio transmitter located 410 m up on the Skydeckof Chicago’s Sears Tower, Lyle drops his hammer that falls all the way to theground below. a) How long will it take for Lyle’s hammer to fall? b) With whatspeed will the hammer hit the pavement? c) How far will the hammer havefallen after 1.50 s when a janitor watches it pass outside an office window?

A-13: On July 31, 1994, Sergey Bubka of the Ukraine broke his own world pole-vaulting record by attaining a height of 6.14 m. a) How long did it take Bubkato return to the ground from the highest part of his vault? b) Describe howthis time compares to the time it took him to go from the ground to thehighest point.

A-14: A Christmas tree ball will break if dropped on a hardwood floor with a speedof 2.0 m/s or more. Holly is decorating her Christmas tree when her cat,Trickor, taps a ball, causing it to fall 15 cm from a tree branch to the floor.Does the ball break?

A-15: Perhaps sometime in the future, NASA will develop a program to land ahuman being on Mars. If you were the first Mars explorer and discovered thatwhen you dropped a hammer it took 0.68 s to fall 0.90 m to the ground, whatwould you calculate for the gravitational acceleration on Mars?

Challenge Exercises for Further Study

B-1: Seth is doing his student driving with the “Give-Me-A-Brake” driving schooland is traveling down the interstate with a speed of 9.0 m/s. Mack is drivinghis “18-wheeler” down the fast lane at 27.0 m/s when he notices Seth 30.0 mahead of him in the right lane. a) If Mack and Seth maintain their speeds, howfar must Mack travel before he catches up to Seth? b) How long will this take?

B-2: At the 2000 summer Olympics in Sydney Australia, the women’s 400-mmedley swimming relay was won by the United States. The four U.S. womenswam the 100.0-m leg of the race with the following average speeds: BarbaraBedford (backstroke) at 1.6289 m/s, Megan Quann (breaststroke) at 1.5085 m/s, Jenny Thompson (fly) at 1.7467 m/s and Dara Torres (freestyle) at1.8737 m/s. a) How far was the team’s final time from the world record time

Motion 13

of 4.028 min. set by the Chinese in 1994? b) Did the American women breakthe world record, or miss it? c) What was the U.S. team’s average speed forthe 400.0-m race?

B-3: In 1945, the Enola Gay, a B-29 bomber, dropped the atomic bomb from a height of 9450 m over Hiroshima, Japan. If the plane carrying the bomb weretraveling with a horizontal velocity of 67.0 m/s, how far horizontally wouldthe bomb have traveled between the point of release and the point where itexploded 513 m above the ground? (To avoid being above the bomb when itexploded, the Enola Gay turned sharply away after the bomb’s release.)

B-4: Pepe, the clown, is jumping on a trampoline as Babette, the tightrope walker,above him suddenly loses her balance and falls off the tightrope straighttoward Pepe. Pepe has just started upward at 15 m/s when Babette begins tofall. Pepe catches her in midair after 1.0 s. a) How far has Babette fallen whenshe is caught by Pepe? b) What is Babette’s velocity at the time of contact? c) What is Pepe’s velocity at the time of contact? d) How far above thetrampoline was Babette before she fell?

B-5: Mr. DeFronzo has just learned that he won the Presidential Award forExcellence in Science Teaching. He runs to the open window and throws hisred marking pen into the air with an initial upward speed of 5.00 m/s. a) Ifthe window is 12.0 m above the ground, what is the velocity of the pen 1.0 safter it is thrown? b) How far has the pen fallen from its starting position after2.0 s? c) How long does it take the pen to hit the ground?

B-6: On October 24, 1901 Annie Edson Taylor, a school teacher from Michigan,became the first person to successfully ride over Niagara Falls in a woodenbarrel. Assume Annie began her journey at Goat Island, 240. m from the falls,where the water current started her down the Niagara River at 8.00 m/s.During her journey, the current reached 15.0 m/s as it carried Annie overHorseshoe Falls, a drop of 51.0 m. How long was Annie’s trip from start tofinish?

14 Motion

2 Vectors and Projectiles

15

2-1 Vectors and ScalarsVocabulary Vector: A quantity with magnitude (size) and direction.

Some examples of vectors are displacement, velocity, acceleration, and force.

Vocabulary Scalar: A quantity with magnitude only.

Some examples of scalars are distance, speed, mass, time, and volume.

Vectors are represented by arrows. They can be added by placing the arrowshead to tail. The arrow that extends from the tail of the first vector to the headof the last vector is called the resultant. It indicates both the magnitude anddirection of the vector sum.

Remember, vectors don’t always have to be in a straight line but may be oriented atangles to each other, such as

Resultant vectors can be determined by anumber of different methods. Here you will solve vector addition exercises both graphically and with vector components.

Graphical addition of vectors: Using a ruler, draw all vectors to scale and connect them headto tail. The resultant is the vector that connectsthe tail of the first vector with the head of thelast. (Hint: Using graph paper makes thismethod even easier!)

Vector Components: Because a vector has both magnitude and direction, you can separate it intohorizontal (or x) and vertical (or y) components.To do this, draw a rectangle with horizontal andvertical sides and a diagonal equal to the vector.Draw arrow heads on one horizontal and onevertical side to make the original vector theresultant of the horizontal and vertical components.

After you have drawn the components, you can find their lengths by usingsimple trigonometry. If you are not familiar with trigonometry or need a quickrefresher, refer to Appendix A.

Solved Examples

Example 1: Every March, the swallows return to San Juan Capistrano, California aftertheir winter in the south. If the swallows fly due north and cover 200 km onthe first day, 300 km on the second day, and 250 km on the third day, draw avector diagram of their trip and find their total displacement for the three-dayjourney.

Solution: Because the swallows continue to fly in the same direction throughout the entire trip,these vectors simply add together. This can beshown by placing the displacement vectors headto tail.

200 km � 300 km � 250 km � 750 km north

Example 2: In the record books, there are men who claim that they have such strong teeththat they can even use them to move cars, trains, and helicopters. Joe Ponderof Love Valley, North Carolina is one such man. Suppose a car pullingforward with a force of 20 000 N was pulled back by a rope that Joe held inhis teeth. Joe pulled the car with a force of 25 000 N. Draw a vector diagramof the situation and find the resultant force.

Solution: In this exercise, the vectors are pointing in opposite directions, sothe situation would look like this.

25 000 N � 20 000 N � 5000 N in the direction Joe is pulling. Strong teeth!


Example 3: If St. Louis Cardinals homerun king, Mark McGwire, hit a baseball due westwith a speed of 50.0 m/s, and the ball encountered a wind that blew it northat 5.00 m/s, what was the resultant velocity of the baseball?

Solution: Begin by drawing a vector diagram of the situation.

Solve using the Pythagorean theorem:

a2 � b2 � c2

(50.0 m/s)2 � (5.00 m/s)2 � c2

c � � 50.2 m/s toward the northwest

For those of you who understand trigonometry, you can find the exact angleat which the ball travels by saying:

tan � � � � 0.100 so tan�1 0.100 � 5.71°

However, don’t worry. If you are not familiar with trigonometry, you cansimply write the answer as 50.2 m/s to the north of west. For a brief review oftrigonometry, see Appendix A.

Example 4: The Maton family begins a vacation trip by driving 700 km west. Then thefamily drives 600 km south, 300 km east, and 400 north. Where will theMatons end up in relation to their starting point? Solve graphically.

Solution: First, draw the appropriate diagram to scale using a relationshipsuch as 1 cm � 1 km, and you will seea space remaining between where theMatons began their trip and wherethey ended. Because you are solvingthis exercise graphically, measure witha ruler the length of the remainingspace and convert your measurementback into km. This is the resultantdisplacement. (Hint: You may find iteasier to use graph paper for yourdrawing so that you can have 1 kmequal to a certain number of squares.)

Answer is 450 km, as measured with a ruler.

5.00 m>s50.0 m>s

opp

adj

22500 m2>s2 � 25.0 m2>s2

Vectors and Projectiles 17

Example 5: Ralph is mowing the back yard with a push mower that he pushes downwardwith a force of 20.0 N at an angle of 30.0° to the horizontal. What are thehorizontal and vertical components of the force exerted by Ralph?

Solution: Begin solving by drawing a diagram of the situation, labeling thehorizontal and vertical components of theforce.

Horizontal component: The hypotenuse inthis exercise is the 20.0-N force. The horizontal component is the one going inthe x direction. This is the side adjacent to the 30.0° angle so you use theequation for the cosine of an angle.

cos � � Fx � F cos � � (20.0 N) cos 30.0° � 17.3 N

Vertical component: Again, the 20.0-N force is the hypotenuse of the triangle.The vertical component is the one going in the y direction. This is the sideopposite the 30.0° angle so you use the equation for the sine of an angle.

sin � � Fy � F sin � � (20.0 N) sin 30.0° � 10.0 N

Practice Exercises

Exercise 1: Some Antarctic explorers heading due south toward the pole travel 50. kmduring the first day. A sudden snow storm slows their progress and theymove only 30. km on the second day. With plenty of rest they travel the final65 km the last day and reach the pole. What was the explorers’ displacement?

Answer:

Fy

F

Fx

F


Exercise 2: Erica and Tory are out fishing on the lake on a hot summer day when theyboth decide to go for a swim. Erica dives off the front of the boat with a forceof 45 N, while Tory dives off the back with a force of 60. N. a) Draw a vectordiagram of the situation. b) Find the resultant force on the boat.

Answer: b.

Exercise 3: Young thoroughbreds are sometimes reluctant to enter the starting gate fortheir first race. Astro Turf is one such horse, and it takes two strong men to gethim set for the race. Derek pulls Astro Turf’s bridle from the front with a forceof 200. N and Dan pushes him from behind with a force of 150. N, while thehorse pushes back against the ground with a force of 300. N. a) Draw a vectordiagram of the situation. b) What is the resultant force on Astro Turf?

Answer: b.

Exercise 4: Shareen finds that when she drives her motorboat upstream she can travelwith a speed of only 8 m/s, while she moves with a speed of 12 m/s whenshe heads downstream. What is the current of the river on which Shareen is traveling?

Answer:


Exercise 5: Rochelle is flying to New York for her big Broadway debut. If the plane headsout of Los Angeles with a velocity of 220. m/s in a northeast direction, relativeto the ground, and encounters a wind blowing head-on at 45 m/s, what is theresultant velocity of the plane, relative to the ground?

Answer:

Exercise 6: While Dexter is on a camping trip with his boy scout troop, the scout leaderhands each boy a compass and map. The directions on Dexter’s map read asfollows: “Walk 500.0 m north, 200.0 m east, 300.0 m south, and 400.0 m west.”If he follows the map, what is Dexter’s displacement? Solve graphically.

Answer:

Exercise 7: Amit flies due east from San Francisco to Washington, D.C., a displacement of5600. km. He then flies from Washington to Boston, a displacement of 900. kmat an angle of 55.0° east of north. What is Amit’s total displacement?

Answer:


216337 km 22 � 1516 km 22

Exercise 8: Marcie shovels snow after a storm by exerting a force of 30.0 N on her shovelat an angle of 60.0° to the vertical. What are the horizontal and verticalcomponents of the force exerted by Marcie?

Answer:

Answer:

Exercise 9: Ivan pulls a sled loaded with logs to his cabin in the woods. If Ivan pulls witha force of 800. N in a direction 20.0° above the horizontal, what are thehorizontal and vertical components of the force exerted by Ivan?

Answer:

Answer:

2-2 Projectile MotionVocabulary Projectile: An object that moves through space acted upon only by the earth’s

gravity.

A projectile may start at a given height and move toward the ground in an arc. For example, picturethe path a rock makes when it is tossed straightout from a cliff.

Horizontal component: Fx � Fo cos � �


A projectile may also start at a given level and then move upward and downwardagain as does a football that has beenthrown.

Regardless of its path, a projectile will always follow these rules:

1. Projectiles always maintain a constant horizontal velocity (neglecting airresistance).

2. Projectiles always experience a constant vertical acceleration of 10.0 m/s2

downward (neglecting air resistance).

3. Horizontal and vertical motion are completely independent of each other.Therefore, the velocity of a projectile can be separated into horizontal andvertical components.

4. For a projectile beginning and ending at the same height, the time it takes torise to its highest point equals the time it takes to fall from the highest pointback to the original position.

5. Objects dropped from a moving vehicle have the same velocity as themoving vehicle.

In order to solve projectile exercises, you must consider horizontal and verticalmotion separately. All of the equations for linear motion in Chapter 1 can beused for projectile motion as well. You don’t need to learn any new equations!

To simplify calculations, the term for initial vertical velocity, vyo, will be leftout of all equations in which an object is projected horizontally. For example,

�dy � vyo�t � ( )g�t2 will be written as �dy � ( )g�t2.

Solved Examples

Example 6: In her physics lab, Melanie rolls a 10-g marble down a ramp and off the tablewith a horizontal velocity of 1.2 m/s. The marble falls in a cup placed 0.51 mfrom the table’s edge. How high is the table?

Solution: The first thing you should notice about projectile exercises is that you do notneed to consider the mass of the objectprojected. Remember, if you ignore airresistance, all bodies fall at exactly the samerate regardless of their mass.

12

12


Before you can find the height of the table, you must first determine how longthe marble is in the air. The horizontal distance traveled equals the constanthorizontal velocity times the travel time.

Given: �dx � 0.51 m Unknown: �t � ?vx � 1.2 m/s Original equation: vx �

Solve: �t � � � 0.43 s

Now that you know the time the marble takes to fall, you can find the verticaldistance it traveled.

Given: g � 10.0 m/s2 Unknown: �dy � ?�t � 0.43 s Original equation: �dy � g�t2

Solve: �dy � (10.0 m/s2)(0.43 s)2 � 0.92 m

Example 7: Bert is standing on a ladder picking apples in his grandfather’s orchard. As hepulls each apple off the tree, he tosses it into a basket that sits on the ground3.0 m below at a horizontal distance of 2.0 m from Bert. How fast must Bertthrow the apples (horizontally) in order for them to land in the basket?

Solution: Before you can find the horizontal component of the velocity, youmust first find the time that the apple is in the air.

Given: �dy � 3.0 m Unknown: �t � ?g � 10.0 m/s2 Original equation: �dy � g�t2

Solve: t � � � 0.77 s

Now that you know the time, you can use it to find the horizontal componentof the velocity.

Given: �dx � 2.0 m Unknown: vx � ?�t � 0.77 s Original equation: �dx � vx �t

Solve: vx � � � 2.6 m/s

Example 8: Emanuel Zacchini, the famous human cannonball of the Ringling Bros. andBarnum & Bailey Circus, was fired out of a cannon with a speed of 24.0 m/sat an angle of 40.0° to the horizontal. If he landed in a net 56.6 m away at thesame height from which he was fired, how long was Zacchini in the air?

Solution: Because Zacchini was in the air for the same amount of timevertically that he was horizontally, you can find his horizontal time and thiswill be the answer. First, you need the horizontal velocity component.

2.0 m0.77 s

¢dx

¢t

B 213.0 m 210.0 m>s2B2¢dy

g

a 12b

a 12b

a 12b

0.51 m1.2 m>s

¢dxvx

¢dx

¢t


cos � � vx � v cos � � (24.0 m/s) cos 40.0° � 18.4 m/s

Now you have the horizontal velocity component and the horizontaldisplacement, so you can find the time.

Given: vx � 18.4 m/s Unknown: �t � ?�dx � 56.6 m Original equation: �dx � vx�t

Solve: �t � � � 3.08 s

Example 9: On May 20, 1999, 37-year old Robbie Knievel, son of famed daredevil EvelKnievel, successfully jumped 69.5 m over a Grand Canyon gorge. Assumingthat he started and landed at the same level and was airborne for 3.66 s, whatheight from his starting point did this daredevil achieve?

Solution: Because 3.66 s is the time for the entire travel through the air,Robbie spent half of this time reaching the height of the jump. The motorcycletook 1.83 s to go up, and another 1.83 s to come down. To find the height themotorcycle achieved, look only at its downward motion as measured from thehighest point.

Given: �t � 1.83 s Unknown: �dy � ?g � 10.0 m/s2 Original equation: �dy � g�t2

Solve: �dy � g�t2 � (10.0 m/s2)(1.83 s)2 � 16.7 m

Practice Exercises

Exercise 10: Billy-Joe stands on the Talahatchee Bridge kicking stones into the water below.a) If Billy-Joe kicks a stone with a horizontal velocity of 3.50 m/s, and it landsin the water a horizontal distance of 5.40 m from where Billy-Joe is standing,what is the height of the bridge? b) If the stone had been kicked harder, howwould this affect the time it would take to fall?

Answer: a.

Answer: b.

a 12ba 1

2b

a 12b

56.6 m18.4 m>s

¢dxvx

vxv


a 12ba 1

2b

Exercise 11: The movie “The Gods Must Be Crazy” begins with a pilot dropping a bottleout of an airplane. It is recovered by a surprised native below, who thinks it isa message from the gods. If the plane from which the bottle was dropped wasflying at an altitude of 500. m, and the bottle lands 400. m horizontally fromthe initial dropping point, how fast was the plane flying when the bottle wasreleased?

Answer:

Exercise 12: Tad drops a cherry pit out the car window 1.0 m above the ground whiletraveling down the road at 18 m/s. a) How far, horizontally, from the initialdropping point will the pit hit the ground? b) Draw a picture of the situation.c) If the car continues to travel at the same speed, where will the car be inrelation to the pit when it lands?

Answer: a.

Answer: c.

Exercise 13: Ferdinand the frog is hopping from lily pad to lily pad in search of a good flyfor lunch. If the lily pads are spaced 2.4 m apart, and Ferdinand jumps with aspeed of 5.0 m/s, taking 0.60 s to go from lily pad to lily pad, at what anglemust Ferdinand make each of his jumps?

Answer:


2211.0 m 2 > 110.0 m>s2 222¢dy>g

221500. m 2 > 110.0 m>s2 222¢dy>g

Exercise 14: At her wedding, Jennifer lines up all the single females in a straight line awayfrom her in preparation for the tossing of the bridal bouquet. She stands Kellyat 1.0 m, Kendra at 1.5 m, Mary at 2.0 m, Kristen at 2.5 m, and Lauren at 3.0 m. Jennifer turns around and tosses the bouquet behind her with a speedof 3.9 m/s at an angle of 50.0° to the horizontal, and it is caught at the sameheight 0.60 s later. a) Who catches the bridal bouquet? b) Who might havecaught it if she had thrown it more slowly?

Answer: a.

Answer: b.

Exercise 15: At a meeting of physics teachers in Montana, the teachers were asked tocalculate where a flour sack would land if dropped from a moving airplane.The plane would be moving horizontally at a constant speed of 60.0 m/s at analtitude of 300. m. a) If one of the physics teachers neglected air resistancewhile making his calculation, how far horizontally from the dropping pointwould he predict the landing? b) Draw a sketch that shows the path the floursack would take as it falls to the ground (from the perspective of an observeron the ground and off to the side.)

Answer: a.

Exercise 16: Jack be nimble, Jack be quick, Jack jumped over the candlestick with avelocity of 5.0 m/s at an angle of 30.0° to the horizontal. Did Jack burn his feeton the 0.25-m-high candle?

Answer:


221300. m 2 > 110.0 m>s2 222¢dy>g


A-1: A flock of Canada geese is flying south for the winter. On the first day thegeese fly due south a distance of 800. km. On the second day they fly backnorth 100. km and pause for a couple of days to graze on a sod farm. The last day the geese continue their journey due south, covering a distance of750. km. a) Draw a vector diagram of the journey and find the totaldisplacement of the geese during this time. b) How does this value differ from the total distance traveled?

A-2: A seal swims toward an inlet with a speed of 5.0 m/s as a current of 1.0 m/sflows in the opposite direction. How long will it take the seal to swim 100. m?

A-3: In Moncton, New Brunswick, each high tide in the Bay of Fundy produces alarge surge of water known as a tidal bore. If a riverbed fills with this flowingwater that travels north with a speed of 1.0 m/s, what is the resultant velocityof a puffin who tries to swim east across the tidal bore with a speed of 4.0 m/s?

A-4: Lynn is driving home from work and finds that there is road constructionbeing done on her favorite route, so she must take a detour. Lynn travels 5 kmnorth, 6 km east, 3 km south, 4 km west, and 2 km south. a) Draw a vectordiagram of the situation. b) What is her displacement? Solve graphically. c) What total distance has Lynn covered?

A-5: Avery sees a UFO out her bedroom window and calls to report it to the police.She says, “The UFO moved 20.0 m east, 10.0 m north, and 30.0 m west beforeit disappeared.” What was the displacement of the UFO while Avery waswatching? Solve graphically.

A-6: Eli finds a map for a buried treasure. It tells him to begin at the old oak andwalk 21 paces due west, 41 paces at an angle 45° south of west, 69 paces duenorth, 20 paces due east, and 50 paces at an angle of 53° south of east. Howfar from the oak tree is the buried treasure? Solve graphically.

A-7: Dwight pulls his sister in her wagon with a force of 65 N at an angle of 50.0°to the vertical. What are the horizontal and vertical components of the forceexerted by Dwight?

A-8: Esther dives off the 3-m springboard and initially bounces up with a velocityof 8.0 m/s at an angle of 80.° to the horizontal. What are the horizontal andvertical components of her velocity?

A-9: In many locations, old abandoned stone quarries have become filled withwater once excavating has been completed. While standing on a 10.0-m-highquarry wall, Clarence tosses a piece of granite into the water below. IfClarence throws the rock horizontally with a velocity of 3.0 m/s, how far outfrom the edge of the cliff will it hit the water?


A-10: While skiing, Ellen encounters an unexpected icy bump, which she leaveshorizontally at 12.0 m/s. How far out, horizontally, from her starting pointwill Ellen land if she drops a distance of 7.00 m in the fall?

A-11: The Essex county sheriff is trying to determine the speed of a car that slid offa small bridge on a snowy New England night and landed in a snow pile 4.00 m below the level of the road. The tire tracks in the snow show that thecar landed 12.0 m measured horizontally from the bridge. How fast was thecar going when it left the road?

A-12: Superman is said to be able to “leap tall buildings in a single bound.” Howhigh a building could Superman jump over if he were to leave the groundwith a speed of 60.0 m/s at an angle of 75.0° to the horizontal?

A-13: Len is running to school and leaping over puddles as he goes. From the edgeof a 1.5-m-long puddle, Len jumps 0.20 m high off the ground with ahorizontal velocity component of 3.0 m/s in an attempt to clear it. Determinewhether or not Len sits in school all day with wet socks on.


B-1: Veronica can swim 3.0 m/s in still water. While trying to swim directly acrossa river from west to east, Veronica is pulled by a current flowing southward at2.0 m/s. a) What is the magnitude of Veronica’s resultant velocity? b) IfVeronica wants to end up exactly across stream from where she began, atwhat angle to the shore must she swim upstream?

B-2: Solve Practice Exercise A-6 using vector components.

B-3: Mubarak jumps and shoots a field goal from the far end of the court into thebasket at the other end, a distance of 27.6 m. The ball is given an initial velocityof 17.1 m/s at an angle of 40.0° to the horizontal from a height of 2.00 m abovethe ground. What is its velocity as it hits the basket 3.00 m off the ground?

B-4: Drew claims that he can throw a dart at a dartboard from a distance of 2.0 mand hit the 5.0-cm-wide bulls-eye if he throws the dart horizontally with aspeed of 15 m/s. He starts the throw at the same height as the top of thebulls-eye. See if Drew is able to hit the bulls-eye by calculating how far hisshot falls from the bulls-eye’s lower edge.

B-5: Caitlin is playing tennis against a wall. She hits the tennis ball from a heightof 0.5 m above the ground with a velocity of 20.0 m/s at an angle of 15.0° tothe horizontal toward the wall that is 6.00 m away. a) How far off the groundis the ball when it hits the wall? b) Is the ball still traveling up or is it on itsway down when it hits the wall?

B-6: From Chapter 1, Exercise B-6, determine how far from the base of NiagaraFalls Annie Taylor landed in her wooden barrel.


3 Forces

29

3-1 Forces and AccelerationVocabulary Force: A push or a pull.

When an unbalanced force is exerted on an object, the object accelerates in thedirection of the force. The acceleration is proportional to the force andinversely proportional to the mass of the object. This is Newton’s second lawand it can be represented with an equation that says

force � (mass)(acceleration) or F � ma

The unbalanced force is called the net force, or resultant of all the forcesacting on the system.

The SI unit for force is the newton, which equals one kilogram meter persecond squared (N � kg �m/s2).

You can think of a newton as being about equivalent to the weight of a stickof butter.

Mass, or the amount of matter in an object, does not change regardless ofwhere an object is located. It is a constant property of any object. However, do not confuse mass with weight! The weight of an object is simply thegravitational force acting on the object. Therefore, if an object is moved awayfrom Earth to a location where g is no longer 10.0 m/s2, the object will nolonger have the same weight as it did on Earth. The equation for weight is just a specific case of F � ma.

weight � (mass)(acceleration due to gravity) or w � mg

Because of its weight, an object pushes against a surface on which it lies. ByNewton’s third law, the surface pushes back on the object. This push, which iscalled the normal force, is always perpendicular to the surface on which theobject rests.

Some of the exercises you will do in this chapter require the use of some basic trigonometry. If you would like a review of trigonometry, refer toAppendix A.

Solved Examples

Example 1: Felicia, the ballet dancer, has a mass of 45.0 kg. a) What is Felicia’s weight onEarth? b) What is Felicia’s mass on Jupiter, where the acceleration due togravity is 25.0 m/s2? c) What is Felicia’s weight on Jupiter?

Solution: a. Felicia’s weight on Earth depends upon the gravitational pull ofthe earth on Felicia’s mass.

Given: m � 45.0 kg Unknown: w � ?g � 10.0 m/s2 Original equation: w � mg

Solve: w � mg � (45.0 kg)(10.0 m/s2) � 450. N

b. The mass of an object remains the same whether the object is on Earth, inspace, or on another planet. Therefore, on Jupiter, Felicia’s mass is still 45.0 kg.

c. The acceleration due to gravity on Jupiter is 25.0 m/s2.


Solve: w � mg � (45.0 kg)(25.0 m/s2) � 1130 N

Since a newton is equivalent to 0.22 pounds, little Felicia would weigh about260 lb on Jupiter. It should be noted, however, that it would be impossible tostand on Jupiter due to its entirely gaseous surface.

Example 2: Butch, the 72.0-kg star quarterback of Belmont High School’s football team,collides with Trask, a stationary left tackle, and is brought to a stop with anacceleration of �20.0 m/s2. a) What force does Trask exert on Butch? b) Whatforce does Butch exert on Trask?

Solution: a. The force depends upon the rate at which Butch’s mass is broughtto rest.

Given: m � 72.0 kg Unknown: F � ?g � �20.0 m/s2 Original equation: F � ma

Solve: F � ma � (72.0 kg)( �20.0 m/s2) � �1440 N

The negative sign in the answer implies that the direction of the force isopposite that of Butch’s original direction of motion.

b. Newton’s third law states that for every action there is an equal andopposite reaction. Therefore, if Trask exerts a force of �1440 N on Butch,Butch will exert the same 1440 N force back on Trask, but in the oppositedirection.

30 Forces

Example 3: A 20-g sparrow flying toward a bird feeder mistakes the pane of glass in awindow for an opening and slams into it with a force of 2.0 N. What is thebird’s acceleration?

Solution: Since the sparrow exerts 2.0 N of force on the window, the windowmust provide �2.0 N back in the opposite direction. Don’t forget to convertgrams into kilograms before beginning.

20 g � 0.02 kg

For a review of unit conversions, see Appendix A.

Given: m � 0.02 kg Unknown: a � ?F � �2.0 N Original equation: F � ma

Solve: a � � � �100 m/s2 (about 10 g’s!)

Therefore, the bird experiences a very rapid negative acceleration, as thewindow brings the bird to a sudden stop. Ouch!

Example 4: A 30.0-g arrow is shot by William Tell through an 8.00-cm-thick apple sittingon top of his son’s head. If the arrow enters the apple at 30.0 m/s andemerges at 25.0 m/s in the same direction, with what force has the appleresisted the arrow?

Solution: First, convert g to kg and cm to m.

30.0 g � 0.0300 kg 8.00 cm � 0.0800 m

Next, find the acceleration of the arrow before finding the force.

Given: vo � 30.0 m/s Unknown: a � ?vf � 25.0 m/s Original equation: vf

2 � vo2 � 2a�d

�d � 0.0800 m

Solve: a � � �

� �1720 m/s2

The negative sign before the answer implies that the apple was causing thearrow to slow down. Now solve for the force exerted by the apple.

Given: m � 0.0300 kg Unknown: F � ?a � �1720 m/s2 Original equation: F � ma

Solve: F � ma � (0.0300 kg)(�1720 m/s2) � �51.6 N

This is the force that the apple exerts on the arrow. It is negative because itsdirection is opposite to the arrow’s direction of motion.

625 m2>s2 � 900. m2>s2

0.160 m125.0 m>s 22 � 130.0 m>s 22

210.0800 m 2vf

2 � vo2

2�d

�2.0 N0.02 kg

Fm

Forces 31

Example 5: Rose is sledding down an ice-covered hill inclined at an angle of 15° with thehorizontal. If Rose and the sled have a combined mass of 54.0 kg, what is theforce pulling them down the hill?

Solution: This exercise is a bit more complex than the preceding examples.Before beginning the solution, look at all the forces on the sled.

a. First, there is the gravitational force, which always acts downward. This isthe weight of the sled. It is labeled mg.

b. The next force to be considered is the normal force. This force always actsperpendicular to the surface, so it pushes against the bottom of the sled. It islabeled FN.

c. The resultant of these forces is a component of the gravitational force thatgoes in the direction of the motion of the sled, or down the slope. It is labeledFd.

You can redraw these three forces as a right triangle. The angle of the slope corresponds to the angle between mg and FN. Nowwith the use of trigonometry, you can solve for the force down the incline, Fd.

Given: m � 54.0 kg Unknown: Fd � ?g � 10.0 m/s2 Original equation: sin � � �� 15°

Solve: Fd � mg sin � � (54.0 kg)(10.0 m/s2) sin 15° � 140 N

Fdmg

opphyp

32 Forces

Practice Exercises

Exercise 1: You can find your own mass in kg with the following information: 1.0 kgweighs about 2.2 lb on Earth. a) What is your mass in kg? b) What is yourweight in newtons?

Answer: a.

Answer: b.

Exercise 2: Gunter the weightlifter can lift a 230.0-kg barbell overhead on Earth. Theacceleration due to gravity on the sun is 274 m/s2. a) Would the barbells beheavier on the sun or on Earth? b) How much (in newtons) would the barbellsweigh on the sun (if it were possible to stand on the sun without melting)?

Answer: a.

Answer: b.

Exercise 3: Sammy Sosa swings at a 0.15 kg baseball and accelerates it at a rate of3.0 � 104 m/s2. How much force does Sosa exert on the ball?

Answer:

Forces 33

Exercise 4: Claudia stubs her toe on the coffee table with a force of 100. N. a) What is theacceleration of Claudia’s 1.80-kg foot? b) What is the acceleration of the tableif it has a mass of 20.0 kg? (Ignore any frictional effects.) c) Why wouldClaudia’s toe hurt less if the table had less mass?

Answer: a.

Answer: b.

Answer: c.

Exercise 5: While chopping down his father’s cherry tree, George discovered that if heswung the axe with a speed of 25 m/s, it would embed itself 2.3 cm into thetree before coming to a stop. a) If the axe head had a mass of 2.5 kg, howmuch force was the tree exerting on the axe head upon impact? b) How muchforce did the axe exert back on the tree?

Answer: a.

Answer: b.

Exercise 6: Carter’s favorite ride at Playland Amusement Park is the rollercoaster. Therollercoaster car and passengers have a combined mass of 1620 kg, and theydescend the first hill at an angle of 45.0° to the horizontal. With what force isthe rollercoaster pulled down the hill?

Answer:

34 Forces

3-2 FrictionVocabulary Friction: The force that acts to oppose the motion between two materials

moving past each other.

There are many types of friction between surfaces. They include

Static friction: The resistance force that must be overcome to start an object inmotion.

Kinetic or sliding friction: The resistance force between two surfaces alreadyin motion.

Rolling friction: The resistance force between a surface and a rolling object.

Fluid friction: The resistance force of a gas or a liquid as an object passesthrough. One example of fluid friction is air resistance.

In this chapter, we will deal only with kinetic or sliding friction.

The force of sliding friction between two surfaces depends on the normalforce pressing the surfaces together, and on the types of surfaces that are incontact with each other. The magnitude of this force is written as

force of sliding friction � (coefficient of sliding friction)(normal force)

or Ff � �FN

If an object is sitting on a horizontal surface, the normal force is equal to theweight of the object. The symbol � (pronounced “mu”) is called the coefficientof sliding friction. A high coefficient of friction (in other words, a largenumber for �) means that the object is not likely to slide easily, while a lowcoefficient of friction (or a small �) is found between very slippery surfaces.Because the coefficient of sliding friction is simply a ratio of the force of slidingfriction to the normal force, it has no units.

Solved Examples

Example 6: Brian is walking through the school cafeteria but does not realize that theperson in front of him has just spilled his glass of chocolate milk. As Brian,who weighs 420 N, steps in the milk, the coefficient of sliding friction betweenBrian and the floor is suddenly reduced to 0.040. What is the force of slidingfriction between Brian and the slippery floor?

Solution: In order to find the force of sliding friction, you need to know thenormal force, or the force the ground exerts upward on Brian. On a horizontal

Forces 35

surface this normal force is equivalent to the object’s weight, which in thiscase is 420 N.

Given: FN � 420 N Unknown: Ff � ?� � 0.040 Original equation: Ff � �FN

Solve: Ff � �FN � (0.040)(420 N) � 17 N

Example 7: While redecorating her apartment, Kitty slowly pushes an 82-kg china cabinetacross the wooden dining room floor, which resists the motion with a force offriction of 320 N. What is the coefficient of sliding friction between the chinacabinet and the floor?

Solution: As in the previous exercise, the normal force is equivalent to theweight of the china cabinet because the cabinet is sitting on a horizontalsurface.

Given: m � 82 kg Unknown: w � ?g � 10.0 m/s2 Original equation: w � mg

Solve: w � mg � (82 kg)(10.0 m/s2) � 820 N so FN is also 820 N.

Given: FN � 820 N Unknown: � � ?Ff � 320 N Original equation: Ff � �FN

Solve: � � � � 0.39 Remember, � has no units!

Example 8: At Sea World, a 900.-kg polar bear slides down a wet slide inclined at an angleof 25.0° to the horizontal. The coefficient of friction between the bear and theslide is 0.0500. What frictional force impedes the bear’s motion down the slide?

Solution: In this example, unlike the previous ones in this section, the polar bear is inclined atan angle to the horizontal so you must find thenormal force on the polar bear by using thecosine of this angle. Remember, the normal force,FN, always acts perpendicular to the surface onwhich the object is moving.

Given: m � 900. kg Unknown: FN � ?g � 10.0 m/s2 Original equation: cos � � �� 25.0°

Solve: FN � mg cos � � (900. kg)(10.0 m/s2) cos 25.0° � 8160 N


Solve: Ff � �FN � (0.0500)(8160 N) � 408 N

FNmg

adj

hyp

320 N820 N

Ff

FN

36 Forces

Practice Exercises

Exercise 7: Unbeknownst to the students, every time the school floors are waxed, Mr.Tracy, the principal, likes to slide down the hallway in his socks. Mr. Tracyweighs 850. N and the coefficient of sliding friction between his socks and thefloor is 0.120. What is the force of friction that opposes Mr. Tracy’s motiondown the hall?

Answer:

Exercise 8: Skye is trying to make her 70.0-kg Saint Bernard go out the back door but thedog refuses to walk. If the coefficient of sliding friction between the dog andthe floor is 0.50, how hard must Skye push in order to move the dog with aconstant speed?

Answer:

Exercise 9: Rather than taking the stairs, Martin gets from the second floor of his house tothe first floor by sliding down the banister that is inclined at an angle of 30.0°to the horizontal. a) If Martin has a mass of 45 kg and the coefficient of slidingfriction between Martin and the banister is 0.20, what is the force of frictionimpeding Martin’s motion down the banister? b) If the banister is madesteeper (inclined at a larger angle), will this have any effect on the force offriction? If so, what?

Answer: a.

Answer: b.

Forces 37

Exercise 10: As Alan is taking a shower, the soap falls out of the soap dish and Alan stepson it with a force of 500 N. If Alan slides forward and the frictional forcebetween the soap and the tub is 50 N, what is the coefficient of frictionbetween these two surfaces?

Answer:

Exercise 11: Howard, the soda jerk at Bea’s diner, slides a 0.60-kg root beer from the end ofthe counter to a thirsty customer. A force of friction of 1.2 N brings the drink toa stop right in front of the customer. a) What is the coefficient of sliding frictionbetween the glass and the counter? b) If the glass encounters a sticky patch onthe counter, will this spot have a higher or lower coefficient of friction?

Answer: a.

Answer: b.

3-3 StaticsVocabulary Statics: The study of forces in equilibrium.

When forces are in equilibrium, all the forces acting on a body are balanced,and the body is not accelerating. In order to solve statics exercises, you muststudy all the forces acting on an object in the horizontal or x-directionseparately from all the forces acting in the vertical or y-direction. This meansthat you must take the horizontal and vertical components of these forces.Because the object is not accelerating, the sum of all the horizontal componentsmust equal zero and the sum of all the vertical components must equal zero.Rules for finding horizontal and vertical components are found in Appendix A.

A few hints: In statics exercises, you may frequently see the term tension.Tension is the force that is exerted by a rope or a wire, or any object that pullson another. It has the same units as any other force.

a) � � Ff/FN � Ff/mg � (1.2 N)/(0.60 kg)(10.0 m/s2) � 0.20

38 Forces

You will notice that there are several exercises here that involve objectshanging from wires. Whenever this situation occurs, the sum of the verticalcomponents of the tension in each wire is equal to the object’s weight. If theobject hangs in the middle of two equal-length wires, the weight is sharedequally by each wire.

Solved Examples

Example 9: Flip, an exhausted gymnast, hangs from a bar by both arms in an effort tocatch his breath. If Flip has a mass of 65.0 kg, what is the tension in each ofFlip’s arms as he hangs in place?

Solution: First find Flip’s weight.


Solve: w � mg � (65.0 kg)(10.0 m/s2) � 650. N

Since Flip is pulling down on the bar with a force of 650. N, his arms must beholding him up by sharing an upward force of 650. N. If each of Flip’s armsshares the force equally, then each must provide a tension of 325 N.

Example 10: At an art auction, Whitney has acquired a painting that now hangs from a nailon her wall, as shown in the figure. If the painting has a mass of 12.6 kg, whatis the tension in each side of the wire supporting the painting?

Solution: The weight of the painting is shared equally by two wires, so each wire must support only half of theweight. However, in this example the wires do not hangvertically, but instead act at an angle with the painting.Therefore, you must use trigonometry to find the actualtension in the wire.


Solve: w � mg � (12.6 kg)(10.0 m/s2) � 126 N

Therefore, each of the wires equally shares 63 N. Call this value Fy, and use trigonometry to find the angle.

cos � � � F � � � � 89 N

Therefore, each wire holds the painting with a tension or force of 89 N.

63 N0.71

63 Ncos 45°

Fy

cos u

Fy

F

adj

hyp

Forces 39

Example 11: Michelle likes to swing on a tire tied to a tree branch in her yard, as in the figure. a) If Michelleand the tire have a combined mass of 82.5 kg, andElwin pulls Michelle back far enough for her tomake an angle of 30.0° with the vertical, what isthe tension in the rope supporting Michelle andthe tire?


Solve: w � mg � (82.5 kg)(10.0 m/s2) � 825 N

First, redraw the forces so that they are connected head to tail in a triangle. This allows you to use rulesof trigonometry to solve for the tension, F.

cos � � �

F � � � � 953 N

Example 12: After returning home from the beach, Samantha hangs her wet 0.20-kg bathing suit in the centerof the 6.0-m-long clothesline to dry. This causesthe clothesline to sag 4.0 cm. What is the tensionin the clothesline?

Solution: First, convert cm to m.

4.0 cm � 0.040 m

Because the bathing suit is hung in the center of the clothesline, the tension ineach side of the line is the same. You must find the downward force on theclothesline, which is simply the weight of the bathing suit.


Solve: w � mg � (0.20 kg)(10.0 m/s2) � 2.0 N

825 N0.866

825 Ncos 30.0°

mg

cos u

mg

F

adj

hyp

40 Forces

Half of this force, 1.0 N, will pull on the left side of the clothesline and half willpull on the right, only because the bathing suit hangs in the middle of the line.

Before you can calculate the tension, F, in the rope, you need to determine the angle theclothesline makes with the horizontal. To do this,use the known distances of 3.0 m and 0.040 m asshown in the diagram. This diagram is not drawnto scale.

tan � � � � � 0.013

Taking the inverse of the tangent gives the angle,

tan�1 0.013 � � so � � 0.74°

The angle is 0.74° and the bathing suit causes the clothesline to pull up with a force of 1.0 N. Nowfind the tension in the line. Again, this is doneusing trigonometry, because the angle and thevertical component of the force are known.

sin � � �

F � � � � 77 N

The tension in each side of the clothesline is 77 N.

Practice Exercises

Exercise 12: While moving out of her dorm room, Bridget carries a 12-kg box to her car,holding it in both arms. a) How much force must be exerted by each of herarms to support the box? b) How will this force change if Bridget holds thebox with only one arm?

Answer: a.

Answer: b.

1.0 N0.013

1.0 Nsin 0.74°

Fy

sin u

Fy

F

opp

hyp

0.040 m3.0 m

dy

dx

opp

adj

(Not drawn to scale)

Forces 41

Exercise 13: A flower pot of mass 4.20 kg is hung above a window by three ropes, eachmaking an angle of 15.0° with the vertical, as shown. What is the tension ineach rope supporting the flower pot?

Answer:

Exercise 14: Luke Skywalker must swing Princess Leia across a large chasm in order toescape the Storm Troopers. If Luke and Leia’s combined mass is 145 kg,calculate the tension in the rope just before Luke and Leia start their swing,when the pair makes an angle of 30.0° with the vertical.

Answer:

Exercise 15: The ACE towing company tows a disabled 1050-kg automobile off the road ata constant speed. If the tow line makes an angle of 10.0° with the vertical asshown, what is the tension in the line supporting the car?

Answer:

F � mg/cos � � (1050 kg)(10.0 m/s2)/cos 10.0° � 10 600 N

42 Forces

13

Exercise 16: Yvette hangs a 2.4-kg bird feeder in the middle of a rope tied between twotrees. The feeder creates a tension of 480 N in each side of the the rope. a) If the two trees are 4.0 m apart, how much will the rope sag in the center?b) If a bird lands on the feeder, will this have any effect on the tension in therope? Explain.

Answer: a.

Answer: b.

Exercise 17: After pulling his car off to the side of the road during a rainstrom, Travis isdismayed to find that the car has become stuck in the mud. Travis ties oneend of a rope to the front of the car and the other end to a tree 12.00 m awayas shown. a) If Travis can exert a force of 610 N on the rope, moving it 1.00 min the direction shown, how much force will the rope exert on the car? b) Whyis this method better than simply tying a rope to the front of the car andpulling the car straight out?

Answer: a.

Answer: b.

Forces 43

3-4 PressureVocabulary Pressure: The force per unit area.

pressure � or P �

The SI unit for pressure is the pascal, which equals one newton per squaremeter (P � N/m2).

It is very easy to confuse pressure with force. While force is a push or a pull,pressure is a push or pull on a certain area. For a given force, the pressure dueto that force is inversely proportional to the area on which the force is exerted.Therefore, if the area of contact is small, the amount of pressure between twosurfaces is much greater than if the force were exerted over a larger area.

For example, place a pencil between the palms of your hands with the pointedend pushing against one palm and the eraser end against the other. As yousqueeze your hands together, you will feel a much more unpleasant sensationat the pencil tip than at the eraser! The eraser has a larger area, so the force isspread out more evenly over the nerve endings of your hand.

Solved Examples

Example 13: Brooke comes home from school and puts her books down on the kitchentable while she goes to grab a snack. The books have a combined weight of 25 N and the area of contact is 0.19 m by 0.24 m. What pressure do the booksapply on the table?

Solution: First, find the area of the surface that is pressing down on the table.

area � length � width � 0.19 m � 0.24 m � 0.046 m2

Given: F � 25 N Unknown: P � ?A � 0.046 m2 Original equation: P �

Solve: P � � � 540

Example 14: A full coffee mug has a mass of 0.60 kg and an empty mug has a mass of 0.30 kg. a) Which mug, the full one or the empty one, applies a greaterpressure on the table? b) If the full mug applies a pressure of 1200. N/m2,what is the area inside a circular ring of coffee left on the table by the bottomof the mug? c) What is the radius of the ring of coffee?

a. The full mug applies more pressure because a larger force is spread over thegiven area.

Nm2

25 N0.046 m2

FA

FA

FA

forcearea

44 Forces

b. The force exerted by the full mug is its weight.

w � mg � (0.60 kg)(10.0 m/s2) � 6.0 N

Given: F � 6.0 N Unknown: A � ?P � 1200. N/m2 Original equation: P �

Solve: A � � � 0.0050 m2

c. To find the radius, use the equation for the area of a circle.

Given: A � 0.0050 m2 Unknown: r � ?� � 3.14 Original equation: A � �r2

Solve: r � � � 0.040 m

Practice Exercises

Exercise 18: a) Which exerts a greater force on a table, a 1.70-kg physics book lying flat onthe table, or a 1.70-kg physics book standing on end on the table? b) Whichapplies a greater pressure? c) If each book measures 0.260 m � 0.210 m �0.040 m, calculate the pressure applied in each of these two drawings.

Answer: a.

Answer: b.

Answer: c.

B0.0050 m2

3.14BA�

6.0 N1200. N>m2

FP

FA

Forces 45

Exercise 19: Miss Culp, a high school English teacher, marches next to Miss Vance, aphysics teacher, in the graduation procession across the football field. Eachwoman has a mass of 60.0 kg, but Miss Culp is wearing spike heels that havean area of 0.40 cm2 while Miss Vance wears wide heels with an area of 6.0 cm2. a) Calculate how much pressure each woman will apply on theground. b) What could Miss Culp do, while she walks, to help her sink lessinto the ground?

Answer: a.

Answer: b.

Exercise 20: Morgan has a mass of 85 kg and is on top of a bed in such a position that shecan apply a pressure of 9530 N/m2 on the mattress. Would you calculate thatMorgan is standing, sitting, or lying on the bed?

Answer:

Exercise 21: Caleb is filling up water balloons for the Physics Olympics balloon tosscompetition. Caleb sets a 0.50-kg spherical water balloon on the kitchen tableand notices that the bottom of the balloon flattens until the pressure on thebottom is reduced to 630 N/m2. a) What is the area of the flat spot on thebottom of the balloon? b) What is the radius of the flat spot?

Answer: a.

Answer: b.

46 Forces

210.0079 m2 2 > 13.14 22A>p


A-1: What is the minimal force a mother must exert to lift her 5.0-kg baby out of itscrib?

A-2: On the moon, the gravity is 1/6 that of Earth. While on the moon, Buzz Aldrincarried on his back a support system that would weigh over 1760 N on Earth.a) What did the backpack weigh on the moon? b) What was its mass on themoon?

A-3: A common malady in runners who run on too hard a surface is shin splints. If a runner’s 7.0-kg leg hits the pavement so that it comes to rest with anacceleration of �200.0 m/s2 on each hit, how much force must the runner’sleg withstand on each step?

A-4: In the district soccer championship finals, Elizabeth kicks a 0.600-kg soccerball with a force of 80.0 N. How much does she accelerate the soccer ball fromrest in the process?

A-5: Barker is unloading 20-kg bottles of water from this delivery truck when oneof the bottles tips over and slides down the truck ramp that is inclined at anangle of 30° to the ground. What amount of force moves the bottle down theramp?

A-6: Sarah, whose mass is 40.0 kg, is on her way to school after a winter stormwhen she accidentally slips on a patch of ice whose coefficient of slidingfriction is 0.060. What force of friction will eventually bring Sarah to a stop?

A-7: In her physics lab, Molly puts a 1.0-kg mass on a 2.0-kg block of wood. Shepulls the combination across another wooden board with a constant speed todetermine the coefficient of sliding friction between the two surfaces. If Mollymust pull with a force of 6.0 N, what coefficient of sliding friction does shecalculate for wood on wood?

A-8: A 1250-kg slippery hippo slides down a mud-covered hill inclined at an angleof 18.0° to the horizontal. a) If the coefficient of sliding friction between thehippo and the mud is 0.0900, what force of friction impedes the hippo’smotion down the hill? b) If the hill were steeper, how would this affect thecoefficient of sliding friction?

A-9: Erma receives a 5.00-kg package in the mail tied with a string that goes around each side of thebox, as shown. If Erma lifts the box by the stringin the center so that each piece of string makes anangle of 45.0° with the vertical, what is thetension in each piece of string?

Forces 47

A-10: To make extra money during the summer, Mr. Garber, a 66.0-kg physicsteacher, paints the outside of houses while sitting on a 4.0-kg plank suspendedby two vertical cables. What is the tension in each of the two cables?

A-11: While camping in Denali National Park in Alaska, a wise camper hangs hispack of food from a rope tied between two trees, to keep the food away fromthe bears. If the 5.0-kg bag of food hangs from the center of a rope that is 3.0 m long, and the rope sags 6.0 cm in the middle, what is the tension in the rope?

A-12: In the figure, a 1240-kg wrecking ball is pulled back with a horizontal force of 5480 N beforebeing swung against the side of a building. a) What angle does the wrecking ball make withthe vertical when it is pulled back? b) What is thetension in the ball’s supporting cable when it is atthis angle?

A-13: What force must you exert on a ball point pen inorder to apply a pressure of 0.067 N/mm2 on apiece of paper, if the ball of the pen has a surfacearea of 1.2 mm2 touching the paper?

A-14: Asad cuts his knee in a fall while chasing a soccer ball. If a 6-N force isexerted on Asad’s knee during the fall, applying a pressure of 1000 N/m2 onan area of his skin, what is the area of the cut that results from the impact?

A-15: The amazing Gambini walks across a 30.0-m-long tightrope high above a 3-ring circus. a) If the 75.0-kgGambini pushes the tightrope down 15.0 cm in thecenter, find the tension in the tightrope. b) If a 10-cm2

area of Gambini’s foot presses on the rope, how muchpressure does Gambini apply on this area?

A-16: In the TV show, The Addams Family, Uncle Fester found it quite comfortable tosleep on a bed of nails. Though this doesn’t sound like the most pleasant wayto take a nap, it is not too painful if many nails are placed fairly close together.a) If Uncle Fester has a mass of 53 kg and his body covers 700 nails, each witha surface area of 1.00 mm2, what is the pressure exerted on his body? b) Whatwould be the pressure if Uncle Fester napped on a bed made of only 1 nail?


Example 15: Linc, the 65.0-kg lifeguard, slides down a water slide that is inclined at anangle of 35.0° to the horizontal, into the community swimming pool. If thecoefficient of friction of the slide is 0.050, what is Linc’s rate of acceleration ashe slides down?

48 Forces

Solution: Start by constructing a triangle showing all the forces acting on the lifeguard.Then find the normal force acting on Lincwhen he is inclined at an angle to thehorizontal. Because the normal force alwaysacts perpendicular to the surface on whichthe object sits, find this force with the use oftrigonometry.

cos � � �

FN � mg cos � � (65.0 kg)(10.0 m/s2) cos 35.0° � 532 N

Now use this normal force to find the force of friction.


Solve: Ff � �FN � (0.050)(532 N) � 27 N

Next, return to the original triangle to find the downward component of theweight, which pulls Linc down the slide.

sin � � �

Fd � mg sin � � (65.0 kg)(10.0 m/s2) sin 35.0° � 373 N

The exercise asks for Linc’s acceleration at the bottom of the slide. Becausefriction opposes Linc’s motion, subtract its effect from Fd. The net force actingon Linc is

Fnet � Fd � Ff � 373 N � 27 N � 346 N

Now solve for the rate of acceleration.

Given: F � 346 N Unknown: a � ?m � 65.0 kg Original equation: F � ma

Solve: a � � � 5.32 m/s2346 N65.0 kg

Fm

Fdmg

opphyp

FNmg

adj

hyp

Forces 49

B-1: Malcolm, the 20.0-kg monkey, hangs from a jungle vine, as shown. a) What is the tension in the segmentof vine labeled AB? b) What is the tension in thesegment of the vine labeled BC?

B-2: Noah is loading the ark and the last animal on boardis a stubborn 1500-kg elephant who refuses to budge.Noah and his family pull the elephant at a constantspeed up the 10° incline with a force of 10 000 N.What is the coefficient of sliding friction between theelephant and the loading platform?

B-3: Blythe lies in a hospital bed with her foot in traction, as shown. How much tensionwill the traction device exert on her foot?

B-4: Madison, whose mass is 35.0 kg, climbs theladder on the slide in her back yard, andslides to the ground at an angle of 30.0° tothe horizontal. If the coefficient of slidingfriction is 0.15, what is Madison’sacceleration down the slide? Ignore theinitial effects of starting friction.

B-5: A chunk of rock of mass 50.0 kg slides down the side of a volcano that slopesup at an angle of 30.0° to the horizontal. If the rock accelerates at a rate of 3.0 m/s2, what is the coefficient of sliding friction between the rock and theside of the volcano?

B-6: While waterskiing behind her father’s boat, Cheryl is pulled at a constantspeed with a force of 164 N by a rope that makes an angle of 10.0° with thehorizontal. If Cheryl has a mass of 65.0 kg, what is the coefficient of slidingfriction between Cheryl and the water?

B-7: Gooluk, the Inuit, is pulling a 62.0-kg sled through the snow on his way homefrom ice fishing. On the back of the sled is his 50.0-kg sack of fishing tackle.The coefficient of sliding friction between the sled and the snow is 0.0700 andthe coefficient of sliding friction between the sled and the sack is 0.100. Whilepulling, the fishing rod sticking out of his sack catches on a tree branch, butGooluk doesn’t notice and keeps walking. What force does Gooluk need toexert to keep the sled moving with a constant speed while the sack is pulledback across it?

50 Forces

4 Momentum

51

4-1 Impulse and MomentumVocabulary Momentum: A measure of how difficult it is to stop a moving object.

momentum � (mass)(velocity) or p � mv

If the momentum of an object is changing, as it is when a force is exerted tostart it or stop it, the change in momentum can be found by looking at thechange in mass and velocity during the interval.

change in momentum � change in [(mass)(velocity)] or �p � �(mv)

For all the exercises in this book, assume that the mass of the object remainsconstant, and consider only the change in velocity, �v, which is equal tovf � vo. Momentum is a vector quantity. Its direction is in the direction of theobject’s velocity.

The SI unit for momentum is the kilogram�meter/second (kg�m/s).

Vocabulary Impulse: The product of the force exerted on an object and the time intervalduring which it acts.

impulse � (force)(elapsed time) or J � F�t

The SI unit for impulse is the newton �second (N�s).

The impulse given to an object is equal to the change in momentum of theobject.

F�t � m�v

The same change in momentum may be the result of a large force exerted fora short time, or a small force exerted for a long time. In other words, impulseis the thing that you do, while change in momentum is the thing that you see.

The units for impulse and momentum are equivalent. Remember, 1 N � 1 kg �m/s2. Therefore, 1 N �s � 1 kg �m/s.

Solved Examples

Example 1: Tiger Woods hits a 0.050-kg golf ball, giving it a speed of 75 m/s. Whatimpulse does he impart to the ball?

Solution: Because the impulse equals the change in momentum, you canreword this exercise to read, “What was the ball’s change in momentum?” It isunderstood that the ball was initially at rest, so its initial speed was 0 m/s.

Given: m � 0.050 kg Unknown: �p � ?�v � 75 m/s Original equation: �p � m�v

Solve: �p � (0.050 kg)(75 m/s) � 3.8 kg �m/s

Example 2: Wayne hits a stationary 0.12-kg hockey puck with a force that lasts for1.0 � 10�2 s and makes the puck shoot across the ice with a speed of 20.0 m/s,scoring a goal for the team. With what force did Wayne hit the puck?

Given: m � 0.12 kg Unknown: F � ?�v � 20.0 m/s Original equation: F�t � m�v�t � 1.0 � 10�2 s

Solve: F � � � 240 kg �m/s2 � 240 N

Example 3: A tennis ball traveling at 10.0 m/s is returned by Venus Williams. It leaves herracket with a speed of 36.0 m/s in the opposite direction from which it came.a) What is the change in momentum of the tennis ball? b) If the 0.060-kg ballis in contact with the racket for 0.020 s, with what average force has Venus hitthe ball?

Solution: In this exercise, the tennis ball is coming toward Venus with a speedof 10.0 m/s in one direction, but she hits it back with a speed of 36.0 m/s inthe opposite direction. Therefore, you must think about velocity vectors andcall one direction positive and the opposite direction negative.

a. Given: vo � �10.0 m/s Unknown: �p � ?vf � 36.0 m/s Original equation: �p � m�v � m(vf � vo)m � 0.060 kg

Solve: �p � m(vf � vo) � (0.060 kg)[36.0 m/s � (�10.0 m/s)] � 2.8 kg �m/s

b. Given: m � 0.060 kg Unknown: F � ?�v � 46.0 m/s Original equation: F�t � m�v�t � 0.020 s

Solve: F � � � 140 N10.060 kg 2 146.0 m>s 2

10.020 s 2m¢v¢t

10.12 kg 2 120.0 m>s 21.0 � 10�2 s

m¢v¢t

52 Momentum

Example 4: To demonstrate his new high-speed camera, Flash performs an experiment inthe physics lab in which he shoots a pellet gun at a pumpkin to record themoment of impact on film. The 1.0-g pellet travels at 100. m/s until it embedsitself 2.0 cm into the pumpkin. What average force does the pumpkin exert tostop the pellet?

Solution: First, convert g to kg and cm to m.

1.0 g � 0.0010 kg 2.0 cm � 0.020 m

Before you can solve for the force in the exercise, you must first know howlong the force is being exerted. Remember, in order to find the time, you mustuse the average velocity, vav.

vav � � � 50.0 m/s

Given: v � 50.0 m/s Unknown: �t � ?�d � 0.020 m Original equation: �d � v�t

Solve: �t � � � 0.00040 s

Now we can solve for the force the pumpkin exerts to stop the pellet.

Given: m � 0.0010 kg Unknown: F � ?�v � 100. m/s Original equation: F�t � m�v�t � 0.0040 s

Solve: F � � � 250 N

Practice Exercises

Exercise 1: On April 15, 1912, the luxury cruiseliner Titanic sank after running into aniceberg. a) What momentum would the 4.23 � 108-kg ship have imparted tothe iceberg if it had hit the iceberg head-on with a speed of 11.6 m/s?(Actually, the impact was a glancing blow.) b) If the captain of the ship hadseen the iceberg a kilometer ahead and had tried to slow down, why wouldthis have been a futile effort?

Answer: a.

Answer: b.

10.0010 kg 2 1100. m>s 210.00040 s 2

m¢v¢t

0.020 m50.0 m>s

¢dv

0 m>s � 100. m>s2

vf � vo

2

Momentum 53

Exercise 2: Auto companies frequently test the safety of automobiles by putting themthrough crash tests to observe the integrity of the passenger compartment. If a 1000.-kg car is sent toward a cement wall with a speed of 14 m/s and theimpact brings it to a stop in 8.00 � 10�2 s, with what average force is itbrought to rest?

Answer:

Exercise 3: Rhonda, who has a mass of 60.0 kg, is riding at 25.0 m/s in her sports carwhen she must suddenly slam on the brakes to avoid hitting a dog crossingthe road. She is wearing her seatbelt, which brings her body to a stop in 0.400 s. a) What average force did the seatbelt exert on her? b) If she had notbeen wearing her seatbelt, and the windshield had stopped her head in1.0 � 10�3 s, what average force would the windshield have exerted on her?c) How many times greater is the stopping force of the windshield than theseatbelt?

Answer: a.

Answer: b.

Answer: c.

Exercise 4: If 270 million people in the United States jumped up in the air simultaneously,pushing off Earth with an average force of 800. N each for a time of 0.10 s,what would happen to the 5.98 � 1024 kg Earth? Show a calculation thatjustifies your answer.

Answer:

54 Momentum

Exercise 5: In Sharkey’s Billiard Academy, Maurice is waiting to make his last shot. Henotices that the cue ball is lined up for a perfect head-on collision, as shown.Each of the balls has a mass of 0.0800 kg and the cue ball comes to a completestop upon making contact with the 8 ball. Suppose Maurice hits the cue ballby exerting a force of 180. N for 5.00 � 10 �3 s, and it knocks head-on into the8 ball. Calculate the resulting velocity of the 8 ball.

Answer:

Exercise 6: During an autumn storm, a 0.012-kg hail stone traveling at 20.0 m/s made a0.20-cm-deep dent in the hood of Darnell’s new car. What average force didthe car exert to stop the damaging hail stone?

Answer:

4-2 Conservation of MomentumAccording to the law of conservation of momentum, the total momentum ina system remains the same if no external forces act on the system. Considerthe two types of collisions that can occur.

Vocabulary Elastic collision: A collision in which objects collide and bounce apart with noenergy loss.

In an elastic collision, because momentum is conserved, the mv before acollision for each of the two objects must equal the mv after the collision foreach of the two objects. This is written as

m1v1o � m2v2o � m1v1f � m2v2f

The subscripts 1 and 2 refer to objects 1 and 2, respectively.

Momentum 55

Vocabulary Inelastic collision: A collision in which objects collide and some mechanicalenergy is transformed into heat energy.

A common kind of inelastic collision is one in which the colliding objects sticktogether, or start out stuck together and then separate. However, in an inelasticcollision the objects need not remain stuck together but may instead deform insome way.

Because momentum is also conserved in an inelastic collision, the mv beforethe collision for each of the two objects must equal the mv after the collisionfor each of the two objects. When objects are stuck together after the collision(assuming mass does not change), this equation becomes

m1v1o � m2v2o � (m1 � m2)vf

where vf is the combined final velocity of the two objects.

Solved Examples

Example 5: Tubby and his twin brother Chubby have a combined mass of 200.0 kg andare zooming along in a 100.0-kg amusement park bumper car at 10.0 m/s.They bump Melinda’s car, which is sitting still. Melinda has a mass of 25.0 kg.After the elastic collision, the twins continue ahead with a speed of 4.12 m/s.How fast is Melinda’s car bumped across the floor?

Solution: Notice that you must add the mass of the bumper car to the mass ofthe riders.

Given: m1 � 300.0 kg Unknown: v2f � ?m2 � 125.0 kg Original equation:v1o � 10.0 m/s m1v1o � m2v2o � m1v1f � m2v2fv2o � 0 m/sv1f � 4.12 m/s

Solve: v2f �

�

� �

� 14.1 m/s

1764 kg #m>s125.0 kg

3000 kg #m>s � 0 kg #m>s � 1236 kg #m>s125.0 kg

1300.0 kg 2 110.0 m>s 2 � 1125.0 kg 2 10 m>s 2 � 1300.0 kg 2 14.12 m>s 2125.0 kg

m1v1o � m2v2o � m1v1fm2

56 Momentum

Example 6: Sometimes the curiosity factor at the scene of a car accident is so great that itactually produces secondary accidents as a result, while people watch to seewhat is going on. If an 800.-kg sports car slows to 13.0 m/s to check out anaccident scene and the 1200.-kg pick-up truck behind him continues travelingat 25.0 m/s, with what velocity will the two move if they lock bumpers after arear-end collision?

Solution: Since the two vehicles lock bumpers, both objects have the samefinal velocity.

Given: m1 � 800. kg Unknown: vf � ?m2 � 1200. kg Original equation:v1o � 13.0 m/s m1v1o � m2v2o � (m1 � m2)vfv2o � 25.0 m/s

Solve: vf � �

� � 20.2 m/s forward

Example 7: Charlotte, a 65.0-kg skin diver, shoots a 2.0-kg spear with a speed of 15 m/s ata fish who darts quickly away without getting hit. How fast does Charlottemove backwards when the spear is shot?

Solution: To start, Charlotte and the spear are together and both are at rest.

Given: m1 � 65.0 kg Unknown: v1f � ?m2 � 2.0 kg Original equation:vo � 0 m/s (m1 � m2)vo � m1v1f � m2v2fv2f � 15.0 m/s

Solve: v1f �

�

� � �0.46 m/s

Remember, the minus sign here is indicating direction. Therefore, Charlottewould travel with a speed of 0.46 m/s in a direction opposite to that of thespear.

�30.kg #m>s65.0 kg

165.0 kg � 2.0 kg 2 10 m>s 2 � 12.0 kg 2 115 m>s 265.0 kg

1m1 � m2 2vo � m2v2fm1

10 400 kg #m>s � 30 000 kg #m>s2000. kg

1800. kg 2 113.0 m>s 2 � 11200. kg 2 125.0 m>s 21800. kg � 1200. kg 2

m1v1o � m2v2o

1m1 � m2 2

Momentum 57

Practice Exercises

Exercise 7: Jamal is at the state fair playing some of the games. At one booth he throws a0.50-kg ball forward with a velocity of 21.0 m/s in order to hit a 0.20-kg bottlesitting on a shelf, and when he makes contact the bottle goes flying forward at30.0 m/s. a) What is the velocity of the ball after it hits the bottle? b) If thebottle were more massive, how would this affect the final velocity of the ball?

Answer: a.

Answer: b.

Exercise 8: Jeanne rolls a 7.0-kg bowling ball down the alley for the league championship.One pin is still standing, and Jeanne hits it head-on with a velocity of 9.0 m/s.The 2.0-kg pin acquires a forward velocity of 14.0 m/s. What is the newvelocity of the bowling ball?

Answer:

Exercise 9: Running at 2.0 m/s, Bruce, the 45.0-kg quarterback, collides with Biff, the90.0-kg tackle, who is traveling at 7.0 m/s in the other direction. Uponcollision, Biff continues to travel forward at 1.0 m/s. How fast is Bruceknocked backwards?

Answer:

58 Momentum

� [(2.0 kg)(0 m/s) � (7.0 kg)(9.0 m/s) � (2.0 kg)(14.0 m/s)]/(7.0 kg)

Exercise 10: Anthony and Sissy are participating in the “Roll-a-Rama” rollerskating dancechampionship. While 75.0-kg Anthony rollerskates backwards at 3.0 m/s,60.0-kg Sissy jumps into his arms with a velocity of 5.0 m/s in the samedirection. a) How fast does the pair roll backwards together? b) If Anthony isskating toward Sissy when she jumps, would their combined final velocity belarger or smaller than your answer to part a? Why?

Answer: a.

Answer: b.

Exercise 11: To test the strength of a retainment wall designed to protect a nuclear reactor,a rocket-propelled F-4 Phantom jet aircraft was crashed head-on into aconcrete barrier at high speed in Sandia, New Mexico on April 19, 1988. TheF-4 phantom had a mass of 19100 kg, while the retainment wall’s mass was469000 kg. The wall sat on a cushion of air that allowed it to move duringimpact. If the wall and F-4 moved together at 8.41 m/s during the collision,what was the initial speed of the F-4 Phantom?

Answer:

Momentum 59

Exercise 12: Valentina, the Russian Cosmonaut, goes outside her ship for a spacewalk, butwhen she is floating 15 m from the ship, her tether catches on a sharp piece ofmetal and is severed. Valentina tosses her 2.0-kg camera away from thespaceship with a speed of 12 m/s. a) How fast will Valentina, whose mass isnow 68 kg, travel toward the spaceship? b) Assuming the spaceship remainsat rest with respect to Valentina, how long will it take her to reach the ship?

Answer: a.

Answer: b.

Exercise 13: A 620.-kg moose stands in the middle of the railroad tracks, frozen by thelights of an oncoming 10 000.-kg locomotive that is traveling at 10.0 m/s. Theengineer sees the moose but is unable to stop the train in time and the mooserides down the track sitting on the cowcatcher. What is the new combinedvelocity of the locomotive and the moose?

Answer:

Exercise 14: Lee is rolling along on her 4.0-kg skateboard with a constant speed of 3.0 m/swhen she jumps off the back and continues forward with a velocity of 2.0 m/srelative to the ground. This causes the skateboard to go flying forward with aspeed of 15.5 m/s relative to the ground. What is Lee’s mass?

Answer:

60 Momentum


A-1: Bernie, whose mass is 70.0 kg, leaves a ski jump with a velocity of 21.0 m/s.What is Bernie’s momentum as he leaves the ski jump?

A-2: Ethel is sitting on a park bench feeding the pigeons when a child’s ball rollstoward her across the grass. Ethel returns the ball to the child by hitting itwith her 2.0-kg pocketbook with a speed of 20 m/s. If the impact lasts for 0.4 s, with what force does Ethel hit the ball?

A-3: When Reggie stepped up to the plate and hit a 0.150-kg fast ball traveling at 36.0 m/s, the impact caused the ball to leave his bat with a velocity of 45.0 m/s in the opposite direction. If the impact lasted for 0.002 s, what forcedid Reggie exert on the baseball?

A-4: The U.S. Army’s parachuting team, the Golden Knights, are on a routinejumping mission over a deserted beach. On a jump, a 65-kg Knight lands onthe beach with a speed of 4.0 m/s, making a 0.20-m deep indentation in thesand. With what average force did the parachuter hit the sand?

A-5: The late news reports the story of a shooting in the city. Investigators thinkthat they have recovered the weapon and they run ballistics tests on the pistolat the firing range. If a 0.050-kg bullet were fired from the handgun with aspeed of 400 m/s and it traveled 0.080 m into the target before coming to rest,what force did the bullet exert on the target?

A-6: About 50 000 years ago, in an area located outside Flagstaff, Arizona, a giant4.5 � 107-kg meteor fell and struck the earth, leaving a 180-m-deep hole nowknown as Barringer crater. If the meteor was traveling at 20 000 m/s uponimpact, with what average force did the meteor hit the earth?

A-7: Astronaut Pam Melroy, history’s third woman space shuttle pilot, flew thespace shuttle Discovery to the International Space Station to completeconstruction in October of 2000. To undock from the space station Pilot Melroyreleased hooks holding the two spacecraft together and the 68 000-kg shuttlepushed away from the space station with the aid of four large springs. a) Ifthe 73 000-kg space station moved back at a speed of 0.50 m/s, how fast andin what direction did the space shuttle move? b) What was the relative speedof the two spacecraft as they separated?

A-8: Tyrrell throws his 0.20-kg football in the living room and knocks over hismother’s 0.80-kg antique vase. After the collision, the football bounces straightback with a speed of 3.9 m/s, while the vase is moving at 2.6 m/s in theopposite direction. a) How fast did Tyrrell throw the football? b) If the footballcontinued to travel at 3.9 m/s in the same direction it was thrown, would thevase have to be more or less massive than 0.80 kg?

A-9: A 300.-kg motorboat is turned off as it approaches a dock and it coasts intoward the dock at 0.50 m/s. Isaac, whose mass is 62.0 kg, jumps off the front

Momentum 61

of the boat with a speed of 3.0 m/s relative to the boat. What is the velocity ofthe boat after Isaac jumps?

A-10: Miguel, the 72.0-kg bullfighter, runs toward an angry bull at a speed of 4.00 m/s. The 550.-kg bull charges toward Miguel at 12.0 m/s and Miguelmust jump on the bull’s back at the last minute to avoid being run over. Whatis the new velocity of Miguel and the bull as they move across the arena?

A-11: A space shuttle astronaut is sent to repair a defective relay in a 600.00-kgsatellite that is traveling in space at 17 000.0 m/s. Suppose the astronaut andhis Manned Maneuvering Unit (MMU) have a mass of 400.00 kg and travel at 17 010.0 m/s toward the satellite. What is the combined velocity when theastronaut grabs hold of the satellite?

A-12: The U.S.S. Constitution, the oldest fully commissioned war ship in the world,is docked in Boston, Massachusetts. Also known as “Old Ironsides” for herseemingly impenetrable hull, the frigate houses 56 pieces of heavy artillery.Mounted on bearings that allow them to recoil at a speed of 1.30 m/s are 20 carronades, each with a mass of 1000. kg. If a carronade fires a 14.5-kgcannonball straight ahead, with what muzzle velocity does the cannonballleave the cannon?


B-1: On a hot sumer afternoon, Keith and Nate are out fishing in their rowboatwhen they decide to jump into the water and go for a swim. Keith, whose massis 65.0 kg, jumps straight off the front of the boat with a speed of 2.00 m/srelative to the boat, while Nate propels his 68.0-kg body simultaneously off theback of the boat at 4.00 m/s relative to the boat. If the 100.-kg boat is initiallytraveling forward at 3.00 m/s, what is its velocity after both boys jump?

B-2: Lilly, whose mass is 45.0 kg, is ice skating with a constant speed of 7.00 m/swhen she hits a rough patch of ice with a coefficient of friction of 0.0800. Howlong will it take before Lilly coasts to a stop?

B-3: In a train yard, train cars are rolled down a long hill in order to link them upwith other cars as shown. A car of mass 4000. kg starts to roll from rest at thetop of a hill 5.0 m high, and inclined at an angle of 5.0° to the horizontal. Thecoefficient of rolling friction between the train and the track is 0.050. Whatvelocity would the car have if it linked up with 3 identical cars sitting on flatground at the bottom of the track? (Hint: The equation for rolling friction isjust like the one for sliding friction.)

62 Momentum

5 Energy and Machines

63

5-1 Work and PowerVocabulary Work: The product of the component of the force exerted on an object in the

direction of displacement and the magnitude of the displacement.

work � (force)(displacement) or W � F�d

The SI unit for work is the joule (J), which equals one newton �meter (N�m).

For maximum work to be done, the object must move in the direction of the force. If the object is moving at an angle to the force, determine thecomponent of the force in the direction of motion. Remember, if the objectdoes not move, or moves perpendicular to the direction of the force, no work has been done.

Vocabulary Power: The rate at which work is done.

power � or P �

The SI unit for power is the watt (W), which equals one joule per second (J/s).One person is more powerful than another if he or she can do more work in agiven amount of time, or can do the same amount of work in less time.

Solved Examples

Example 1: Bud, a very large man of mass 130 kg, stands on a pogo stick. How muchwork is done as Bud compresses the spring of the pogo stick 0.50 m?

Solution: First, find Bud’s weight, which is the force with which hecompresses the pogo stick spring.

Given: m � 130 kg Unknown: w � ?g � 10.0 m/s2 Original equation: w � mg

Solve: w � mg � (130 kg)(10.0 m/s2) � 1300 N

Now use this weight to solve for the work done to compress the spring.

W¢t

workelapsed time

Given: F � 1300 N Unknown: W � ?�d � 0.050 m Original equation: W � F�d

Solve: W � F�d � (1300 N)(0.050 m) � 65 J

Don’t get confused here by the two W’s you see in this example. The w inw � mg means weight while the W in W � F�d means work. There are manyways to tell them apart, the most important of which is to understand howthey are used in the context of the exercise. Also, the units used for each arequite different: weight is measured in newtons, and work is measured injoules. Last of all, weight is a vector and work is a scalar.

Example 2: After finishing her physics homework, Sherita pulls her 50.0-kg body out ofthe living room chair and climbs up the 5.0-m-high flight of stairs to herbedroom. How much work does Sherita do in ascending the stairs?

Solution: First find Sherita’s weight. Her muscles exert a force to carry herweight up the stairs.


Solve: w � mg � (50.0kg)(10.0 m/s2) � 500. N

Now use Sherita’s weight (or force) to determine the amount of work done. Itis important to note that when you are solving for the work done, you needknow only the displacement of the body moved. The number of stairs climbedor their steepness is irrelevant. All that is important is the change in position.

Given: F � 500. N Unknown: W � ?�d � 5.0 m Original equation: W � F�d

Solve: W � F�d � (500. N)(5.0 m) � 2500 J

Example 3: In the previous example, Sherita slowly ascends the stairs, taking 10.0 s to gofrom bottom to top. The next evening, in a rush to catch her favorite TV show,she runs up the stairs in 3.0 s. a) On which night does Sherita do more work?b) On which night does Sherita generate more power?

a) Sherita does the same amount of work on both nights because the force sheexerts and her displacement are the same each time.

b) Sherita’s power output varies because the time taken to do the sameamount of work varies.

First night:

Given: W � 2500 J Unknown: P � ?�t � 10.0 s Original equation: P �

W¢t


Solve: P � � � 250 W

Second night:

Given: W � 2500 J Unknown: P � ?�t � 3.0 s Original equation: P �

Solve: P � � � 830 W

Sherita generates more power on the second night.

Practice Exercises

Exercise 1: On his way off to college, Russell drags his suitcase 15.0 m from the door ofhis house to the car at a constant speed with a horizontal force of 95.0 N. a) How much work does Russell do to overcome the force of friction? b) If thefloor has just been waxed, does he have to do more work or less work tomove the suitcase? Explain.

Answer: a.

Answer: b.

Exercise 2: Katie, a 30.0-kg child, climbs a tree to rescue her cat who is afraid to jump 8.0 m to the ground. How much work does Katie do in order to reach the cat?

Answer:

W � F�d � mg�d � (30.0 kg)(10.0 m/s2)(8.0 m) � 2400 J

2500 J3.0 s

W¢t

W¢t

2500 J10.0 s

W¢t

Energy and Machines 65

Exercise 3: Marissa does 3.2 J of work to lower the window shade in her bedroom adistance of 0.8 m. How much force must Marrisa exert on the window shade?

Answer:

Exercise 4: Atlas and Hercules, two carnival sideshow strong men, each lift 200.-kgbarbells 2.00 m off the ground. Atlas lifts his barbells in 1.00 s and Herculeslifts his in 3.00 s. a) Which strong man does more work? b) Calculate whichman is more powerful.

Answer: a.

Answer: b.

5-2 Energy

Potential and Kinetic Energy

Vocabulary Energy: The ability to do work.

There are many different types of energy. This chapter will focus on onlymechanical energy, or the energy related to position (potential energy) andmotion (kinetic energy).

Vocabulary Potential Energy: Energy of position, or stored energy.

An object gains gravitational potential energy when it is lifted from one levelto a higher level. Therefore, we generally refer to the change in potentialenergy or �PE, which is proportional to the change in height, �h.


� gravitational potential energy � (mass)(acceleration due to gravity)(� height)

or �PE � mg�h

It is important to remember that gravitational potential energy relies onlyupon the vertical change in height, �h, and not upon the path taken.

In addition to gravitational potential energy, there are other forms of storedenergy. For example, when a bow is pulled back and before it is released, theenergy in the bow is equal to the work done to deform it. This stored orpotential energy is written as �PE � F�d. Springs possess elastic potentialenergy when they are displaced from the equilibrium position. The equationfor elastic potential energy will not be used in this chapter.

Vocabulary Kinetic Energy: Energy of motion.

The kinetic energy of an object varies with the square of the speed.

kinetic energy � (mass)(speed)2 or KE � mv2

The SI unit for energy is the joule. Notice that this is the same unit used forwork. When work is done on an object, energy is transformed from one formto another. The sum of the changes in potential, kinetic, and heat energy isequal to the work done on the object. Mechanical energy is transformed intoheat energy when work is done to overcome friction.

Conservation of Energy

According to the law of conservation of energy, energy cannot be created ordestroyed. The total amount of mechanical energy in a system remainsconstant if no work is done by any force other than gravity.

In an isolated system where there are no mechanical energy losses due tofriction

�KE � �PE

In other words, all the kinetic and potential energy before an interactionequals all the kinetic and potential energy after the interaction.

KEo � PEo � KEf � PEf or mvo2 � mgho � mvf

2 � mghf

As a reminder, the terms with the subscript o are the initial conditions, whilethose with the subscript f are final conditions.

a 12ba 1

2b

a 12ba 1

2b


Solved Examples

Example 4: Legend has it that Isaac Newton “discovered” gravity when an apple fell froma tree and hit him on the head. If a 0.20-kg apple fell 7.0 m before hittingNewton, what was its change in PE during the fall?

Solution: For a given object, the change in PE depends only upon the changein position. The apple does not need to fall all the way to the ground toexperience an energy change.

Given: m � 0.20 kg Unknown: �PE � ?g � 10.0 m/s2 Original equation: �PE � mg�h

�h � 7.0 m

Solve: �PE � mg�h � (0.20 kg)(10.0 m/s2)(7.0 m) � 14 J

Example 5: A greyhound at a race track can run at a speed of 16.0 m/s. What is the KE ofa 20.0-kg greyhound as it crosses the finish line?

Given: m � 20.0 kg Unknown: KE � ?v � 16.0 m/s Original equation: KE � mv2

Solve: KE � mv2 � (20.0 kg)(16.0 m/s)2 � 2560 J

Example 6: In a wild shot, Bo flings a pool ball of mass m off a 0.68-m-high pool table,and the ball hits the floor with a speed of 6.0 m/s. How fast was the ballmoving when it left the pool table? (Use the law of conservation of energy.)

Given: vf � 6.0 m/s Unknown: vo � ?g � 10.0 m/s2 Original equation: �KE � �PE

ho � 0.68 mhf � 0 m

Solve: KEo � PEo � KEf � PEf or mvo2 � mgho � mvf

2 � mghf

Notice that mass is contained in each of these equations. Therefore, it cancelsout and does not need to be included in the calculation.

vo � �

�

� � 4.7 m/sR18 m2>s2 � 6.8 m2>s2

12

ba 1

2b 16.0 m>s 22 � 110.0 m>s2 2 10 m 2 � 110.0 m>s2 2 10.68 m 2

12

ba 1

2b vf

2 � ghf � gho

12b

a 12bmvf

2 � mghf � mgho

a 12bm

a 12ba 1

2b

a 12ba 1

2b

a 12b


Example 7: Frank, a San Francisco hot dog vender, has fallen asleep on the job. When anearthquake strikes, his 300-kg hot-dog cart rolls down Nob Hill and reachespoint A at a speed of 8.00 m/s. How fast is the hot-dog cart going at point Bwhen Frank finally wakes up and starts to run after it?

Solution: Because mass is contained in each of these equations, it cancels outand does not need to be included in the calculation. Also, the inclination ofthe hill makes no difference. All that matters is the change in height.

Given: vo � 8.00 m/s Unknown: vf � ?g � 10.0 m/s2 Original equation: �KE � �PE

ho � 50.0 mhf � 30.0 m

Solve: KEo � PEo � KEf � PEf or mvo2 � mgho � mvf

2 � mghf

vf � �

�

�

� � 21.5 m/s2464 m2>s2

R32.0 m2>s2 � 500. m2>s2 � 300. m2>s2

12

ba1

2b 18.00 m>s 22 � 110.0 m>s2 2 150.0 m 2 � 110.0 m>s2 2 130.0 m 2

12

ba 1

2b vo

2 � gho � ghf

12b

a 12bmvo

2 � mgho � mghf

a 12bm

a 12ba 1

2b


Practice Exercises

Exercise 5: It is said that Galileo dropped objects off the Leaning Tower of Pisa todetermine whether heavy or light objects fall faster. If Galileo had dropped a5.0-kg cannon ball to the ground from a height of 12 m, what would havebeen the change in PE of the cannon ball?

Answer:

Exercise 6: The 2000 Belmont Stakes winner, Commendable, ran the horse race at anaverage speed of 15.98 m/s. If Commendable and jockey Pat Day had acombined mass of 550.0 kg, what was their KE as they crossed the finish line?

Answer:

Exercise 7: Brittany is changing the tire of her car on a steep hill 20.0 m high. She tripsand drops the 10.0-kg spare tire, which rolls down the hill with an initialspeed of 2.00 m/s. What is the speed of the tire at the top of the next hill,which is 5.00 m high? (Ignore the effects of rotation KE and friction.)

Answer:


212.00 m>s 22 � 2110.0 m>s2 2 3 120.0 m 2 � 15.00 m 2 42vo

2 � 2g1ho � hf 2

Exercise 8: A Mexican jumping bean jumps with the aid of a small worm that lives insidethe bean. a) If a bean of mass 2.0 g jumps 1.0 cm from your hand into the air,how much potential energy has it gained in reaching its highest point. b) What is its speed as the bean lands back in the palm of your hand?

Answer: a.

Answer: b.

Exercise 9: A 500.-kg pig is standing at the top of a muddy hill on a rainy day. The hill is100.0 m long with a vertical drop of 30.0 m. The pig slips and begins to slidedown the hill. What is the pig’s speed at the bottom of the hill? Use the law ofconservation of energy.

Answer:

Exercise 10: While on the moon, the Apollo astronauts enjoyed the effects of a gravitymuch smaller than that on Earth. If Neil Armstrong jumped up on the moonwith an initial speed of 1.51 m/s to a height of 0.700 m, what amount ofgravitational acceleration did he experience?

Answer:


2212.0 � 10�4 J 2 > 10.0020 kg 222¢PE>m

22110.0 m>s2 2 130.0 m 222g¢h

a 12ba 1

2b

5-3 Machines and EfficiencyVocabulary Machine: A device that helps do work by changing the magnitude or

direction of the applied force.

Three common machines are the lever, pulley, and incline.

lever pulley incline

In an ideal situation, where frictional forces are negligible, work input equalswork output.

Fin�din � Fout�dout

However, situations are never ideal. The actual mechanical advantage, orAMA, of the machine is a ratio of the magnitude of the force out (resistance)to the magnitude of the force in (effort).

actual mechanical advantage � or AMA �

On the other hand, the theoretical or ideal mechanical advantage, IMA, isbased only on the geometry of the system and does not take frictional effectsinto account.

ideal mechanical advantage �

or IMA �

Because no machine is perfect and because you will always get out less workthan you put in, you need to consider the efficiency of the machine that youare using. The more efficient the machine, the greater work output you willget for your work input. The efficiency will always be less than 100%.

¢din

¢dout

distance in (effort distance)

distance out (resistance distance)

Fout

Fin

force out (resistance)

force in (effort)


Vocabulary Efficiency: The ratio of the work output to the work input.

efficiency � � �

Efficiency has no units and is usually expressed as a percent.

Solved Examples

Example 8: A crate of bananas weighing 3000. N is shipped from South America to NewYork, where it is unloaded by a dock worker who lifts the crate by pullingwith a force of 200. N on the rope of a pulley system. What is the actualmechanical advantage of the pulley system?

Given: Fout � 3000. N Unknown: AMA � ?Fin � 200. N Original equation: AMA �

Solve: AMA � � � 15.0

The pulley exerts 15.0 times more force on the crate than the dock workerexerts to pull the rope. Notice that mechanical advantage has no units.

Example 9: Two clowns, of mass 50.0 kg and 70.0 kg respectively, are in a circus actperforming a stunt with a trampoline and a seesaw. The smaller clown standson the lower end of the seesaw while the larger clown jumps from thetrampoline onto the raised side of the seesaw, propelling his friend into theair. a) what is the ideal mechanical advantage of the seesaw? b) If the largerclown exerts a force of 850. N on the seesaw as he jumps, how much force isexerted on the smaller clown?

a. The seesaw acts as a lever with the fulcrum 0.80 m from the left side. Theideal mechanical advantage is found by comparing the two distances.

Given: �din � 2.40 m Unknown: IMA � ?�dout � 0.80 m Original equation: IMA �

Solve: IMA � � � 3.02.40 m0.80 m

¢din

¢dout

¢din

¢dout

3000. N200. N

Fout

Fin

Fout

Fin

AMAIMA

Fout¢dout

Fin¢din

work output

work input


b. To answer this question, assume that the seesaw is 100% efficient and thework out equals the work in (which is highly unlikely!).

Given: Fin � 850. N Unknown: Fout � ?�din � 2.40 m Original equation: Fin�din � Fout�dout

�dout � 0.80 m

Solve: Fout � � � 2550 N

Example 10: A jackscrew with a handle 30.0 cm long is used to lift a car sitting on the jack.The car rises 2.0 cm for every full turn of the handle. What is the idealmechanical advantage of the jack?

Solution: For a screw, IMA � � where 2�r is the circumference of

the circle through which the handle turns, and height, �h, refers to theamount the jack (and hence the automobile) is raised.

Given: r � 30.0 cm Unknown: IMA � ?�h � 2.0 cm Original equation: IMA �

Solve: IMA � � � � 94

Example 11: Jack and Jill went up the hill to fetch a pail of water. At the well, Jill used aforce of 20.0 N to turn a crank handle of radius 0.400 m that rotated an axle ofradius 0.100 m, so she could raise a 60.0-N bucket of water. a) What is theideal mechanical advantage of the wheel? b) What is the actual mechanicaladvantage of the wheel? c) What is the efficiency of the wheel?

Solution: Since the crank handle and the axle both turn in a circle, �din � 2�rc(where rc is the radius of the crank handle) and �dout � 2�ra (where ra is theradius of the axle).

a. Given: rc � 0.400 m Unknown: IMA � ?ra � 0.100 m Original equation: IMA �

Solve: IMA � � � � 4.00

b. The force on the bucket of water is Fout and the force exerted by Jill is Fin.

Given: Fout � 60.0 N Unknown: AMA � ?Fin � 20.0 N Original equation: AMA �

Solve: AMA � � � 3.0060.0 N20.0 N

Fout

Fin

Fout

Fin

2p10.400 m 22p10.100 m 2

2prc

2pra

¢din

¢dout

¢din

¢dout

2p130.0 cm 22.0 cm

2pr¢h

¢din

¢dout

¢din

¢dout

2pr¢h

¢din

¢dout

1850. N 2 12.40 m 20.80 m

Fin¢din

¢dout


c. Given: AMA � 3.00 Unknown: Eff � ?IMA � 4.00 Original equation: Eff �

Solve: Eff � � � 0.750 � 75.0%

Example 12: Clyde, a stubborn 3500-N mule, refuses to walk into the barn, so FarmerMacDonald must drag him up a 5.0-m ramp to his stall, which stands 0.50 mabove ground level. a) What is the ideal mechanical advantage of the ramp?b) If Farmer MacDonald needs to exert a 450-N force on the mule to drag himup the ramp with a constant speed, what is the actual mechanical advantageof the ramp? c) What is the efficiency of the ramp?

Solution: For a ramp, ramp length is �din and ramp height is �dout.

a. Given: �din � 5.0 m Unknown: IMA � ?�dout � 0.50 m Original equation: IMA �

Solve: IMA � � � 10.

b. Given: Fout � 3500 N Unknown: AMA � ?Fin � 450 N Original equation: AMA �

Solve: AMA � � � 7.8

c. Given: IMA � 10. Unknown: Eff � ?AMA � 7.8 Original equation: Eff �

Solve: Eff � � � 0.78 � 78%

Practice Exercises

Exercise 11: Cathy, a 460-N actress playing Peter Pan, is hoisted above the stage in order to“fly” by a stagehand pulling with a force of 60. N on a rope wrapped around apulley system. What is the actual mechanical advantage of the pulley system?

Answer:

7.810.

AMAIMA

AMAIMA

3500 N450 N

Fout

Fin

Fout

Fin

5.0 m0.50 m

¢din

¢dout

¢din

¢dout

3.004.00

AMAIMA

AMAIMA


Exercise 12: A windmill uses sails blown by the wind to turn an axle that allows a grind-stone to grind corn into meal with a force of 90. N. The windmill has sails ofradius 6.0 m blown by a wind that exerts a force of 15 N on the sails, and theaxle of the grindstone has a radius of 0.50 m. a) What is the ideal mechanicaladvantage of the wheel? b) What is the actual mechanical advantage of thewheel? c) What is the efficiency of the wheel?

Answer: a.

Answer: b.

Answer: c.

Exercise 13: Winnie, a waitress, holds in one hand a 5.0-N tray stacked with twelve 3.5-Ndishes. The length of her arm from her hand to her elbow is 30.0 cm and herbiceps muscle exerts a force 5.0 cm from her elbow, which acts as a fulcrum.How much force must her biceps exert to allow her to hold the tray?

Answer:

Exercise 14: When building the pyramids, the ancient Egyptians were able to raise largestones to very great heights by using inclines. If an incline has an idealmechanical advantage of 4.00 and the pyramid is 15.0 m tall, how much of an angle would the incline need in order for the Egyptian builder to reach the top?

Answer:


Exercise 15: The Ramseys are moving to a new town, so they have called in the ACMEmoving company to take care of their furniture. Debbie, one of the movers,slides the Ramseys’ 2200-N china cabinet up a 6.0-m-long ramp to the movingvan, which stands 1.0 m off the ground. a) What is the ideal mechanicaladvantage of the incline? b) If Debbie must exert a 500.-N force to move thechina cabinet up the ramp with a constant speed, what is the actualmechanical advantage of the ramp? c) What is the efficiency of the ramp?

Answer: a.

Answer: b.

Answer: c.


A-1: On a ski weekend in Colorado, Bob, whose mass is 75.0 kg, skis down a hillthat is inclined at an angle of 15.0° to the horizontal and has a vertical rise of25.0 m. How much work is done by gravity on Bob as he goes down the hill?

A-2: A pile driver is a device used to drive stakes into the ground. While buildinga fence, Adam drops a pile driver of mass 3000. kg through a vertical distanceof 8.0 m. The pile driver is opposed by a resisting force of 5.0 � 106 N. Howfar is the stake driven into the ground on the first stroke?

A-3: At Six Flags New England in Agawam, Massachusetts, a ride called theCyclone is a giant roller coaster that ascends a 34.1-m hill and then drops 21.9 m before ascending the next hill. The train of cars has a mass of 4727 kg.a) How much work is required to get an empty train of cars from the groundto the top of the first hill? b) What power must be generated to bring the trainto the top of the first hill in 30.0 s? c) How much PE is converted into KE fromthe top of the first hill to the bottom of the 21.9-m drop?

A-4: A flea gains 1.0 � 10�7 J of PE jumping up to a height of 0.030 m from a dog’sback. What is the mass of the flea?


A-5: At target practice, Diana holds her bow and pulls the arrow back a distance of0.30 m by exerting an average force of 40.0 N. What is the potential energystored in the bow the moment before the arrow is released?

A-6: The coyote, whose mass is 20.0 kg, is chasing the roadrunner when the coyote accidentally runs offthe edge of a cliff and plummets to the ground30.0 m below. What force does the ground exerton the coyote as he makes a coyote-shaped dent0.420 m deep in the ground?

A-7: A 0.080-kg robin, perched on a power line 6.0 m above the ground, swoopsdown to snatch a worm from the ground and then returns to an 8.0-m-hightree branch with his catch. a) By how much did the bird’s PE increase in itstrip from the power line to the tree branch? b) How would your answer havechanged if the bird had flown around a bit before landing on the tree branch?

A-8: Blackie, a cat whose mass is 5.45 kg, is napping on top of the refrigeratorwhen he rolls over and falls. Blackie has a KE of 85.5 J just before he lands onhis feet on the floor. How tall is the refrigerator?

A-9: Calories measure energy we get from food, and one dietary Calorie is equal to4187 J. The average food energy intake for human beings is 2000. Calories/day.Assume you have a mass of 55.0 kg and you want to burn off all the Caloriesyou consume in one day. How high a mountain would you have to climb to doso? (Note: This calculation ignores the large amount of energy the bodycontinually loses to heat.)

A-10: From a height of 2.15 m above the floor of Boston’s Fleet Center, forward PaulPierce tosses a shot straight up next to the basketball hoop with a KE of 5.40 J.If his regulation-size basketball has a mass of 0.600 kg, will his shot go as highas the 3.04-m hoop? Use the law of conservation of energy.

A-11: Mr. Macintosh, a computer technician, uses a screwdriver with a handle ofradius 1.2 cm to remove a screw in the back of a computer. The screw movesout 0.20 cm on each complete turn. What is the ideal mechanical advantage ofthe screwdriver?

A-12: Tom’s favorite pastime is fishing. a) How much work is required for Tom toreel in a 10.0-kg bluefish from the water’s surface to the deck of a fishing boat,5.20 m above the water, if the reel of his fishing pole is 85.0% efficient? b) IfTom applies a force of 15 N to the reel’s crank handle, what is the actualmechanical advantage of the fishing pole? c) What is the ideal mechanicaladvantage of the fishing pole?

A-13: A nutcracker 16 cm long is used to crack open a Brazil nut that is placed 12 cm from where your hand is squeezing the nutcracker. What is the idealmechanical advantage of the nutcracker?



B-1: A 5.00-N salmon swims 20.0 m upstream against a current that provides aresistance of 1.50 N. This portion of the stream rises at an angle of 10.0° withrespect to the horizontal. a) How much work is done by the salmon againstthe current? b) What is the gain in PE by the salmon? c) What is the total workthat must be done by the salmon? d) If the salmon takes 40.0 s to swim thedistance, what power does it exert in doing so?

B-2: A 30-kg shopping cart full of groceries sitting at the top of a 2.0-m hill beginsto roll until it hits a stump at the bottom of the hill. Upon impact, a 0.25-kgcan of peaches flies horizontally out of the shopping cart and hits a parked carwith an average force of 490 N. How deep a dent is made in the car?

B-3: Using her snowmobile, Midge pulls a 60.0-kg skier up a ski slope inclined atan angle of 12.0° to the horizontal. The snowmobile exerts a force of 200. Nparallel to the hill. If the coefficient of friction between the skis and the snowis 0.120, how fast is the skier moving after he has been pulled for 100.0 mstarting from rest? (Ignore the effects of the static friction that must beovercome to initially start him in motion.) Use the law of conservation ofenergy.

B-4: Jose, whose mass is 45.0 kg, is riding his 5.0-kg skateboard down the sidewalkwith a constant speed of 6.0 m/s when he rolls across a 10.0-m-long patch ofsand on the pavement. The sand provides a force of friction of 6.0 N. What isJose’s speed as he emerges from the sandy section?

B-5: Eben lifts an engine out of his Volkswagen with the help of a winch thatallows him to raise the engine 0.020 m for every 0.90 m he pulls on the cable.Eben expends 1000. J of energy to lift the 800.-N engine 0.50 m. a) What is theefficiency of the winch? b) What is the ideal mechanical advantage of thewinch? c) What is the actual mechanical advantage of the winch? d) Whatforce does Eben exert to lift the engine?


6 Circular Motion

81

6-1 Centripetal Acceleration and Force

Period, Frequency, and Speed

Vocabulary Period: The time it takes for one full rotation or revolution of an object.

Vocabulary Frequency: The number of rotations or revolutions per unit time.

Period and frequency are reciprocals of each other. In other words,

T � and f �

Since period is a measure of time, its SI unit is the second, while the unit forfrequency is the reciprocal of this, or 1/second. Another way of writing1/second is with the unit hertz (Hz).

When an object spins in a circle, the distance it travels in one revolution is thecircumference of the circle, 2�r. The time it takes for one revolution is theperiod, T. Therefore,

speed � or v �

where v is called the linear or tangential speed because at any given time, thevelocity is tangent to the circle as shown in the diagram. Although the velocityis constant in magnitude (speed), it is always changing direction.

2prT

2�(radius)period

1T

1f

Centripetal Acceleration and Centripetal Force

An object can move around in a circle with a constant speed yet still beaccelerating because its direction is constantly changing. This acceleration,which is always directed in toward the center of the circle, is calledcentripetal acceleration. The magnitude of this acceleration is written as

centripetal acceleration � or ac �

If a mass is being accelerated toward the center of a circle, it must be actedupon by an unbalanced force that gives it this acceleration. This force, calledthe centripetal force, is always directed inward toward the center of the circle.The magnitude of this force is written as

centripetal force � (mass)(centripetal acceleration)

or Fc � mac �

The units for centripetal acceleration and centripetal force are m/s2 and N,respectively.

Solved Examples

Example 1: After closing a deal with a client, Kent leans back in his swivel chair and spinsaround with a frequency of 0.5 Hz. What is Kent’s period of spin?

Given: f � 0.5 Hz Unknown: T � ?Original equation: T �

Solve: T � � � 2 s

Example 2: Curtis’ favorite disco record has a scratch 12 cm from the center that makesthe record skip 45 times each minute. What is the linear speed of the scratchas it turns?

Solution: The record makes 45 revolutions every 60. seconds, so find theperiod of the record first.

T � � 1.3 s

Given: r � 12 cm Unknown: v � ?T � 1.3 s Original equation: v �

Solve: v � � � 58 cm/s2p112 cm 2

1.3 s2prT

2prT

60. s45 rev

10.5 Hz

1f

1f

mv2

r

v2

r1linear speed 22

radius

82 Circular Motion

Example 3: Missy’s favorite ride at the Topsfield Fair is the rotor, which has a radius of 4.0 m. Theride takes 2.0 s to make one full revolution.a) What is Missy’s linear speed on the rotor?b) What is Missy’s centripetal accelerationon the rotor?

Solution: The ride takes 2.0 s to make onefull revolution, so the period is 2.0 s.

a. Given: r � 4.0 m Unknown: v � ?T � 2.0 s Original equation: v �

Solve: v � � � 13 m/s

b. Given: v � 13 m/s Unknown: ac � ?r � 4.0 m Original equation: ac �

Solve: ac � � � 42 m/s2

Example 4: Captain Chip, the pilot of a 60 500-kg jet plane, is told that he must remain ina holding pattern over the airport until it is his turn to land. If Captain Chipflies his plane in a circle whose radius is 50.0 km once every 30.0 min, whatcentripetal force must the air exert against the wings to keep the planemoving in a circle?

Solution: First, convert km to m and min to s.

50.0 km � 5.00 � 104 m 30.0 min � 1.80 � 103 s

Before solving for the centripetal force, find the speed of the airplane.

Given: T � 1.80 � 103 s Unknown: v � ?r � 5.00 � 104 m Original equation: v �

Solve: v � � � 175 m/s

Use this speed to solve for the centripetal force.

Given: m � 60 500 kg Unknown: Fc � ?v � 175 m/s Original equation: Fc �r � 5.00 � 104 m

Solve: Fc � � � 3.71 � 104 N160 500 kg 2 1175 m>s 22

5.00 � 104 mmv2

r

mv2

r

2p15.00 � 104 m 21.80 � 103 s

2prT

2prT

113 m>s 224.0 m

v2

r

v2

r

2p14.0 m 22.0 s

2prT

2prT

Circular Motion 83

Practice Exercises

Exercise 1: Marianne puts her favorite Backstreet Boys disc in her CD player. If it spinswith a frequency of 1800 revolutions per minute, what is the period of spin ofthe compact disc?

Answer:

Exercise 2: Hamlet, a hamster, runs on his exercise wheel, which turns around once every0.5 s. What is the frequency of the wheel?

Answer:

Exercise 3: A sock stuck to the inside of the clothes dryer spins around the drum onceevery 2.0 s at a distance of 0.50 m from the center of the drum. a) What is thesock’s linear speed? b) If the drum were twice as wide but continued to turnwith the same frequency, would the linear speed of a sock stuck to the insidebe faster than, slower than, or the same speed as your answer to part a?

Answer: a.

Answer: b.

Exercise 4: What is the radius of an automobile tire that turns with a frequency of 11 Hzand has a linear speed of 20.0 m/s?

Answer:

84 Circular Motion

Exercise 5: Luigi twirls a round piece of pizza dough overhead with a frequency of 60 revolutions per minute. a) Find the linear speed of a stray piece ofpepperoni stuck on the dough 10. cm from the pizza’s center. b) In whatdirection will the pepperoni move if it flies off while the pizza is spinning?Explain the reason for your answer.

Answer: a.

Answer: b.

Exercise 6: Earth turns on its axis approximately once every 24 hours. The radius of Earthis 6.38 � 106 m. a) If some astronomical catastrophe suddenly brought Earthto a screeching halt (a physical impossibility as far as we know), with whatspeed would Earth’s inhabitants who live at the equator go flying off Earth’ssurface? b) Because Earth is solid, it must turn with the same frequencyeverywhere on its surface. Compare your linear speed at the equator to yourlinear speed while standing near one of the poles.

Answer: a.

Answer: b.

Exercise 7: Jessica is riding on a merry-go-round on an outer horse that sits at a distanceof 8.0 m from the center of the ride. Jessica’s sister, Julie, is on an inner horselocated 6.0 m from the ride’s center. The merry-go-round turns around onceevery 40.0 s. a) Explain which girl is moving with the greater linear speed. b) What is the centripetal acceleration of Julie and her horse?

Answer: a.

Answer: b.

Circular Motion 85

Exercise 8: A cement mixer of radius 2.5 m turns with a frequency of 0.020 Hz. What isthe centripetal acceleration of a small piece of dried cement stuck to the insidewall of the mixer?

Answer:

Exercise 9: A popular trick of many physics teachers is to swing a pail of water around ina vertical circle fast enough so that the water doesn’t spill out when the pail isupside down. If Mr. Lowell’s arm is 0.60 m long, what is the minimum speedwith which he can swing the pail so that the water doesn’t spill out at the topof the path?

Answer:

Exercise 10: To test their stamina, astronauts are subjected to many rigorous physical testsbefore they fly in space. One such test involves spinning the astronauts in adevice called a centrifuge that subjects them to accelerations far greater thangravity. With what linear speed would an astronaut have to spin in order toexperience an acceleration of 3 g’s at a radius of 10.0 m? (1 g � 10.0 m/s2)

Answer:

86 Circular Motion

2110.0 m>s2 2 10.60 m 22acr

2130.0 m>s2 2 110.0 m 22acr

Exercise 11: At the Fermilab particle accelerator in Batavia, Illinois, protons are acceleratedby electromagnets around a circular chamber of 1.00-km radius to speeds nearthe speed of light before colliding with a target to produce enormous amountsof energy. If a proton is traveling at 10% the speed of light, how muchcentripetal force is exerted by the electromagnets? (Hint: The speed of light is3.00 � 108 m/s, mp � 1.67 � 10�27 kg)

Answer:

Exercise 12: Roxanne is making a strawberry milkshake in her blender. A tiny, 0.0050-kgstrawberry is rapidly spun around the inside of the container with a speed of14.0 m/s, held by a centripetal force of 10.0 N. What is the radius of theblender at this location?

Answer:

6-2 TorqueVocabulary Torque: A measurement of the tendency of a force to produce a rotation about

an axis.

torque � perpendicular force � lever arm or � � F � d

The lever arm, d, is the distance from the pivot point, or fulcrum, to the pointwhere the component of the force perpendicular to the lever arm is beingexerted. The longer the lever arm, the larger the torque. This is why it is easierto loosen a tight screw with a long wrench than with your hand or a shortpair of tweezers.

If a torque causes a counterclockwise rotation of an object around the fulcrum,it is positive. If the torque causes a clockwise rotation of an object around the

Circular Motion 87

fulcrum, it is negative. This convention works even if the object remainsbalanced and the torques just attempt to cause a rotation.

The SI unit for torque is the newton �meter (N�m). However, unlike work,which is measured in the same unit, torque is not a form of energy and is notequivalent to a joule.

In most of the exercises in this book, all the torques are balanced. Forexample, if two people are sitting on either side of a seesaw and they want toremain level, they can position themselves so that all the torques on one sideof the seesaw equal all the torques on the other side. The total torque on asystem equals the sum of all the individual torques, or

� � (F1 � d1) � (F2 � d2) � . . .

The . . . means that there may be more than only two torques acting on asystem at any one time. Keep in mind that when an object is balanced, all thetorques must also balance. Therefore, the total torque, �, is zero.

Vocabulary Center of Gravity: The point on any object that acts like the place at which allthe weight is concentrated.

The weight of an object, which acts as if it is concentrated at the center ofgravity, is one of the forces that can cause it to rotate. The weight produces atorque if the object is not supported at its center of gravity.

Solved Examples

Example 5: Ned tightens a bolt in his car engine by exerting 12 N of force on his wrenchat a distance of 0.40 m from the fulcrum. How much torque must Nedproduce to turn the bolt?

Given: F � 12 N Unknown: � � ?d � 0.40 m Original equation: � � F � d

Solve: � � F � d � (12 N)(0.40 m) � 4.8 N �m

Example 6: Mabel and Maude are seesawing on the school playground and decide to seeif they can move to the correct location to make the seesaw balance. Mabelweighs 400. N and she sits 2.00 m from the fulcrum of the seesaw. Whereshould 450.-N Maude sit to balance the seesaw?

Solution: It helps to draw a diagram of the situation to allow yourself tovisualize what is happening.

88 Circular Motion

Given: F1 � 400. N Unknown: d2 � ?F2 � 450. N Original equation:d1 � 2.00 m � � (F1 � d1) � (F2 � d2)

Solve: If 400.-N Mabel makes the seesaw turn in a counterclockwise direction,then 450.-N Maude makes the seesaw turn in a clockwise direction. Therefore,� � (F1 � d1) � �(F2 � d2). If the seesaw is balanced, then � � 0 and theequation becomes � � (F1 � d1) � �(F2 � d2) � 0, or (F1 � d1) � (F2 � d2).Therefore,

d2 � � � 1.78 m from the fulcrum.

Practice Exercises

Exercise 13: A water faucet is turned on when a force of 2.0 N is exerted on the handle, ata distance of 0.060 m from the pivot point. How much torque must beproduced to turn the handle?

Answer:

Exercise 14: Nancy, whose mass is 60.0 kg, is working at a construction site and she sitsdown for a bite to eat at noon. If Nancy sits on the very end of a 3.00-m-longplank pivoted in the middle on a saw horse, how much torque must her co-worker provide on the other end of the plank in order to keep Nancy fromfalling on the ground?

Answer:

1400. N 2 12.00 m 2450. N

1F1 � d1 2F2

Circular Motion 89

Exercise 15: Barry carries his tray of food to his favorite cafeteria table for lunch. The 0.50-m-long tray has a mass of 0.20 kg and holds a 0.40-kg plate of food 0.20 m from the right edge. Barry holds the tray by the left edge with onehand, using his thumb as the fulcrum, and pushes up 0.10 m from the fulcrum with his finger tips. How much upward force must his finger tipsexert to keep the tray level? b) How might Barry make the tray easier to carryif he still chooses to use only one hand?

Answer: a.

Answer: b.

Exercise 16: Soon-Yi is building a mobile to hang over her baby’s crib. She hangs a 0.020-kg toy sailboat 0.010 m from the left end and a 0.015-kg toy truck 0.20 mfrom the right end of a bar 0.50 m long. If the lever arm itself has negligiblemass, where must the support string be placed so that the arm balances?

Answer:

Exercise 17: Orin and Anita, two paramedics, rush a 60.0-kg man from the scene of anaccident to a waiting ambulance, carrying him on a uniform 3.00-kg stretcherheld by the ends. The stretcher is 2.60 m long and the man’s center of mass is1.00 m from Anita. How much force must Orin and Anita each exert to keepthe man horizontal?

Answer:

90 Circular Motion

6-3 Moment of Inertia and Angular MomentumVocabulary Moment of Inertia: The resistance of an object to changes in its rotational

motion.

The equation for the moment of inertia varies depending upon the shape ofthe rotating object. For an object rotating around an axis at a distance r,

moment of inertia � (mass)(radius)2 or I � mr2

The SI unit for moment of inertia is the kilogram�meter squared (kg�m2).

Other moments of inertia can be found in your textbook, and are summarizedas follows.

hoop rotating about its center: I � mr2

hoop rotating about its diameter: I � ( )mr2

solid cylinder: I � ( )mr2

stick rotating about its center of gravity: I � ( )m�2

stick rotating about its end: I � ( )m�2

solid sphere rotating about its center of gravity: I � ( )mr2

Newton’s first law says that inertia is the tendency of an object to stay at restor remain in motion in a straight line with a constant speed unless acted uponby an unbalanced force. Similarly, an object that is rotating tends to continuespinning at a constant rate unless an unbalanced force acts to alter thatrotation. This is called the rotational inertia.

Think of moment of inertia as being the rotational equivalent of the term“mass.” Just as inertia is greater for a greater mass, rotational inertia is greaterfor a greater moment of inertia.

Vocabulary Angular Momentum: The measure of how difficult it is to stop a rotatingobject.

angular momentum � (mass)(velocity)(radius) or L � mvr

The SI unit for angular momentum is the kilogram �meter squared per second(kg�m2/s).

Think of angular momentum as being the rotational equivalent of linear momentum. Just as linear momentum is the product of the mass and the velocity, angular momentum is the product of the mass and the velocityfor an object rotating at a distance r from the axis.

25

13

112

12

12

Circular Motion 91

Momentum is conserved when no outside forces are acting. Similarly, angularmomentum is conserved when no outside torques are acting. A spinning iceskater has angular momentum. When the skater pulls her arms in (decreasingher radius of spin), she spins faster (increasing her velocity). Doing soconserves her angular momentum.

Solved Examples

Example 7: On the Wheel of Fortune game show, a contestant spins the 15.0-kg wheel thathas a radius of 1.40 m. What is the moment of inertia of this disk-shapedwheel?

Solution: A disk is a thin cylinder, so the moment of inertia of a disk is thesame as that of a cylinder.

Given: m � 15.0 kg Unknown: I � ?r � 1.40 m Original equation: I � ( )mr2

Solve: I � ( )(15.0 kg)(1.40 m)2 � 14.7 kg �m2

Example 8: Trish is twirling her 0.60-m majorette’s baton that has a mass of 0.40 kg. Whatis the moment of inertia of the baton as it spins about its center of gravity?

Given: m � 0.40 kg Unknown: I � ?� � 0.60 m Original equation: I � ( )m�2

Solve: I � ( )m�2 � ( )(0.40 kg)(0.60 m)2 � 0.072 kg �m2

Example 9: At Wellesley College in Massachusetts there is a favorite tradition called hoop rolling. In their capsand gowns, seniors roll wooden hoops in a racein which the winner is said to be the first in theclass to marry. Hilary rolls her 0.2-kg hoop acrossthe finish line. The moment of inertia of the hoopis 0.032 kg �m2. What is the radius of the hoop?

Given: m � 0.2 kg Unknown: r � ?I � 0.032 kg �m2 Original equation: I � mr2

Solve: r � � � � 0.4 m

Example 10: Jupiter orbits the sun with a speed of 2079 m/s at an average distance of 71 398 000 m. a) If Jupiter has a mass of 1.90 � 1027 kg, what is its angularmomentum as it orbits?

20.16 m2B0.032 kg #m2

0.2 kgB Im

12

12

12

12

12

92 Circular Motion

Given: m � 1.90 � 1027 kg Unknown: L � ?v � 2079 m/s Original equation: L � mvrr � 71 398 000 m

Solve: L � mvr � (1.90 � 1027 kg)(2079 m/s)(71 398 000 m) � 2.82 � 1038 kg�m2/s

Practice Exercises

Exercise 18: Veanna is in Las Vegas waiting for her number to be called at the roulettewheel, a large 3.0-kg disk of radius 0.60 m. What is the moment of inertia ofthe wheel?

Answer:

Exercise 19: Earth has a mass of 5.98 � 1024 kg and a radius of 6.38 � 106 m. What is themoment of inertia of Earth as it turns on its axis?

Answer:

Exercise 20: Olga, the 50.0-kg gymnast, swings her 1.6-m-long body around a bar by heroutstretched arms. a) What is Olga’s moment of inertia? b) If Olga were topull in her legs, thereby cutting her body length in half, how would thischange her moment of inertia? (Assume her mass is evenly distributed allalong her body.)

Answer: a.

Answer: b.

Circular Motion 93

14

a 12ba 1

2b

a 25ba 2

5b

14

a 13ba 1

3b

Exercise 21: Priya removes her 0.012-kg, 0.60-cm-diameter wedding band and spins it onthe coffee table on its edge. What is the moment of inertia of the ring?

Answer:

Exercise 22: Hickory dickory dock, the 20.0-g mouse ran up the clock, and took turnsriding on the 0.20-m-long second hand, the 0.20-m-long minute hand, and the0.10-m-long hour hand. What was the angular momentum of the mouse oneach of the three hands?

Answer:

Answer:

Answer:

Exercise 23: In a physics experiment, Ingrid, the ice skater, spins around in the rink at 1.2 m/s with each of her arms stretched out 0.70 m from the center of herbody. In each hand she holds a 1.0-kg mass. If angular momentum isconserved, how fast will Ingrid begin to spin if she pulls her arms to aposition 0.15 m from the center of her body?

Answer:


A-1: In the Biblical tale of David and Goliath, the giant is slain when David hits himwith a rock that he has spun around overhead in a sling. If the rock is spunwith a frequency of 100 revolutions per minute, what is the rock’s period?

94 Circular Motion

a 12ba 1

2b

A-2: Ashton the ant is crawling on the still blade of a ceiling fan when the fan is turned on,causing Ashton to go for a ride. If Ashton sitson the fan blade at a distance of 0.80 m fromthe center of the fan and turns with afrequency of 1.2 Hz, a) how fast does Ashtonspin? b) If Ashton slips off the spinning fan,describe the path he will take.

A-3: In “Rumpelstiltskin,” the miller’s daughter is spinning straw into gold on aspinning wheel that turns at a speed of 7.5 m/s, making one revolution every0.50 s. How long is a strand of gold that makes one complete turn around thewheel?

A-4: A 3.20-kg hawk circles overhead in search of prey. a) If the hawk circles onceevery 10.0 s in a circle 12.0 m in radius, what is the linear speed of the hawk?b) What centripetal force allows him to remain in a circle? c) What is providingthe centripetal force?

A-5: Sasha’s favorite ride at the fair is the Ferris wheel that has a radius of 7.0 m. a) If the ride takes 20.0 s to make one full revolution, what is the linear speedof the wheel? b) What centripetal force will the ride exert on Sasha’s 50.0-kgbody? c) Does Sasha feel as if she is being pulled in or out by the ride? d) Explain the difference between what she feels and what is really happeningat the top and bottom of the wheel.

A-6: In order for Sasha (in A-5) to feel weightless at the top of the ride, a) at whatlinear speed must the Ferris wheel turn? b) At this speed, how much will sheappear to weigh at the bottom of the Ferris wheel?

A-7: Earth orbits the sun approximately once every 365.25 days at an averagedistance of about 1.5 � 1011 m. The mass of Earth is 5.98 � 1024kg. a) What isthe centripetal acceleration of Earth? b) What is the centripetal force of the sunon Earth? c) What is the centripetal force of Earth on the sun? d) If this forceexists between the sun and Earth, does this mean that Earth is “falling into”the sun? Explain.

A-8: Most doorknobs are placed on the side of the door opposite the hinges insteadof in the center of the door. a) Why is this so? b) If a torque of 1.2 N�m isrequired to open a door, how much force must be exerted on a doorknob 0.76 m from the hinges compared to a doorknob in the middle of the door,0.38 m from the hinges?

A-9: Priscilla is working out in the gym with a 2.00-kg mass that she holds in onehand and gradually lifts up and down. a) Will Priscilla find it easier to lift themass if she pivots her arm at the shoulder or at the elbow? b) If Priscilla’s arm is 0.60 m long from her shoulder to her palm and 0.28 m long from herelbow to her palm, how much torque must she produce in each case to lift the weight?

Circular Motion 95

A-10: Leif and Paige are rearranging the heights of their movable bookshelves; theyremove one of the 2.00-kg, 0.60-m-long shelves by the two of them holdingopposite ends. A 5.00-kg stack of books is piled up on the shelf 0.20 m fromLeif. How much force must Leif and Paige each exert to hold the shelf level?

A-11: Brewster hits a 0.30-kg pool ball across the pool table and sinks it in the sidepocket. If the pool ball has a radius of 3.5 cm, what is its moment of inertia asit rolls?

A-12: Rocky, a raccoon, squeezes into a 0.60-m-diameter cylindrical trash can to find a late-night snack. However,the can tips over and begins to roll. If Rocky and the canhave a combined mass of 40.0 kg, what is the moment ofinertia of the system?

A-13: Mieko sharpens a knife on a grinding wheel whoseangular momentum is 27 kg�m2/s. The 5.0-kg wheel has aradius of 0.30 m. What is the linear speed of the wheel?

Challenge Problems for Further Study

B-1: The “Bake-a-Lite” Cake Company truck is on its way to deliver a birthdaycake for the MacKenzie party when it rounds a curve of radius 20.0 m at aspeed of 12 m/s. What coefficient of friction is needed between the cake panand the truck in order to keep the pan from slipping?

B-2: On his way home from the office, Steven’s car rounds an unbanked curve thathas a radius of 100 m. If the coefficient of friction between the tires and theroad is 0.40, what is the fastest speed at which the car can round this curvewithout risking an accident?

B-3: Pretending to be Tarzan, 50.0-kg Zach swings from the end of a 5.0-m-longrope attached to a tree branch. The tree branch will break if subjected to aforce greater than 750 N. What is the maximum speed with which Zach canswing in order to avoid breaking the branch?

B-4: Hanging in front of the office of Lewis Skeirik, D.D.M., is a sign that weighs120 N and is suspended at the end of a 0.80-m-long support beam that weighs10.0 N, as shown. What is the tension in a supporting wire that holds the signat an angle of 20.0°?

96 Circular Motion

7 Law of Universal Gravitation

97

7-1 Gravitational ForceVocabulary Law of Universal Gravitation: Every particle attracts every other particle

with a force that is proportional to the mass of the particles and inverselyproportional to the square of the distance between them.

F �

The sign � means “proportional to.” To make an equation out of the abovesituation, insert a quantity called the universal constant of gravitation, G.

G � 6.67 � 10�11 N �m2/kg2

Now the magnitude of this gravitational force can be represented as

Force �

or F �

Like all other forces, the gravitational force of attraction between two objectsis measured in newtons.

Solved Examples

Example 1: The gravitational force of attraction between Earth and the sun is 1.6 � 1023 N.What would this force have been if Earth were twice as massive?

Solution: The gravitational force of attraction between two bodies isproportional to the mass of each of the two bodies. As one mass increases, the gravitational force between the two bodies increases proportionally.Therefore, if Earth’s mass were doubled, the gravitational force between thesun and Earth would double as well.

Therefore, F � 2Fo � 2(1.6 � 1023 N) � 3.2 � 1023 N

GmMd2

1universal constant of gravitation 2 1mass 1 2 1mass 2 21distance 22

mMd2

Example 2: The gravitational force of attraction between Earth and the sun is 1.6 � 1023 N.What would this gravitational force have been if Earth had formed twice asfar away from the sun?

Solution: The gravitational force of attraction between two bodies is inverselyproportional to the square of the distance between them. In this case, if thedistance is twice as great, the force between Earth and the sun would be 1/4as much.

Therefore, F � or F � � � 4.0 � 1022 N

Example 3: Oliver, whose mass is 65 kg, and Olivia, whose mass is 45 kg, sit 2.0 m apartin their physics classroom. a) What is the force of gravitational attractionbetween Oliver and Olivia? b) Why don’t Oliver and Olivia drift toward each other?

a) Given: mOliver � 65 kg Unknown: F � ?MOlivia � 45 kg Original equation: F �

d � 2.0 mG � 6.67 � 10�11 Nm2/kg2

Solve: F � � � 4.9 � 10�8 N

b) Because the gravitational force of Earth is much greater than the forceOliver and Olivia exert on each other.

Practice Exercises

Exercise 1: When Royce was 10 years old, he had a mass of 30 kg. By the time he was 16 years old, his massincreased to 60 kg. How much larger is thegravitational force between Royce and Earth atage 16 compared to age 10?

Answer:

16.67 � 10�11 Nm2>kg2 2 165 kg 2 145 kg 212.0 m 22

GmMd2

GmMd2

11.6 � 1023 N 24

Fo

41d2


Exercise 2: If John Glenn weighed 640 N on Earth’s surface, a) how much would he haveweighed if his Mercury spacecraft had (hypothetically) remained at twice thedistance from the center of Earth? b) Why is it said that an astronaut is nevertruly “weightless?”

Answer: a.

Answer: b.

Exercise 3: Mr. Gewanter, whose mass is 60.0 kg, is doing a physics demonstration in thefront of the classroom. a) How much gravitational force does he exert on 55.0-kg Martha in the front row, 1.50 m away? b) How does this compare towhat he exerts on 65.0-kg Lester, 4.00 m away in the back row?

Answer: a.

Answer: b.

Exercise 4: Astrologers claim that your personality traits are determined by the positionsof the planets in relation to you at birth. Scientists argue that these gravitationaleffects are so small that they are totally insignificant. Compare the gravitationalattraction between you and Mars to the gravitational attraction between youand your 70.0-kg doctor at the moment of your birth, if the doctor stands 0.500 m away. (Note: MM � 6.42 � 1023 kg, dE to M � 7.83 � 1010 m. This is theaverage distance between Earth and Mars. This distance varies as the twoplanets orbit the sun.)

Answer:

Answer:

Law of Universal Gravitation 99

Exercise 5: Our galaxy, the Milky Way, contains approximately 4.0 � 1011 stars with anaverage mass of 2.0 � 1030 kg each. How far away is the Milky Way from ournearest neighbor, the Andromeda Galaxy, if Andromeda contains roughly thesame number of stars and attracts the Milky Way with a gravitational force of2.4 � 1030 N?

Answer:

Exercise 6: Tides are created by the gravitational attraction of the sun and moon on Earth. Calculate the net force pulling on Earth during a) a new moon, b) a full moon, c) a first quarter moon. The diagram is intended to help yourunderstanding of the situation but is not drawn to scale. (mM � 7.35 � 1022 kg,mE � 5.98 � 1024 kg, mS � 1.99 � 1030 kg, dE�M � 3.84 � 108 m,dE�S � 1.50 � 1011 m)

Answer: a.

Answer: b.

Answer: c.


216.67 � 10�11 N #m2>kg2 2 3 12.0 � 1030 kg 2 14.0 � 1011 2 42> 12.4 � 1030 N 22GmM > F

213.53 � 1022 N 22 � 11.99 � 1020 N 22

7-2 Gravitational AccelerationYou can use the law of universal gravitation to find the gravitationalacceleration, g, of any body if you know that body’s mass and radius. Forexample, let’s look at the situation on Earth. The weight of an object onEarth’s surface is equal to the gravitational force between that object andEarth:

mg �

The m on the left represents the mass of an object, such as a human being. Them on the right side of the equation stands for this same mass, so the termcancels out of the equation. The M on the right represents the mass of Earth or other celestial body on which the person is standing. The d in thedenominator is equal to the radius of the celestial body. So the equationbecomes

g �

In this equation, g is the acceleration due to gravity on the celestial body inquestion. On Earth you already know that this value is 10.0 m/s2.

Solved Examples

Example 4: Temba is standing in the lunch line 6.38 � 106 m from the center of Earth.Earth’s mass is 5.98 � 1024 kg. a) What is the acceleration due to gravity? b) When Temba eats his lunch and his mass increases, does this change theacceleration due to gravity?

a. Given: M � 5.98 � 1024 kg Unknown: g � ?d � 6.38 � 106 m Original equation: g �G � 6.67 � 10�11 N �m2/kg2

Solve: g � � � 9.80 m/s2

b. No, his acceleration due to gravity does not change because it is notdependent on his mass.

Example 5: The sun has a mass that is 333 000 times Earth’s mass and a radius 109 timesEarth’s radius. What is the acceleration due to gravity on the sun?

Solution: One way to solve this exercise is to actually multiply the givenvalues by the mass and radius of Earth. However, there is an easier and much

16.67 � 10�11 N #m2>kg2 2 15.98 � 1024 kg 216.38 � 106 m 22

GMd2

GMd2

GMd2

GmMd2


neater way to come up with the correct answer. By working with ratios, youcan find an answer without any information about Earth.

Given: MS � 333 000 ME Unknown: g � ?G � 6.67 � 10�11 N �m2/kg2 Original equation: g �ds � 109 dE

Solve: Set up the above equation as a ratio of sun to Earth before substitutingnumbers.

�

Simplifying gives � � � � 28.0

Therefore, gS � 28.0 gE so the acceleration due to gravity on the sun is 28.0 times what it is on Earth. In other words, it is 28.0 times 10.0 m/s2, or280. m/s2.

Practice Exercises

Exercise 7: In The Little Prince, the Prince visits a small asteroid called B612. If asteroidB612 has a radius of only 20.0 m and a mass of 1.00 � 104 kg, what is theacceleration due to gravity on asteroid B612?

Answer:

Exercise 8: In Exercise 5 in the previous section, what is the Andromeda Galaxy’sacceleration rate toward the Milky Way?

Answer:

1333 000 21109 22

1333 000 ME 2 1dE2 2

1ME 2 1109dE 22MSdE

2

MEdS2

gSgE

GMS

dS2

GME

dE2

gSgE

GMd2


Exercise 9: Black holes are suspected when a visible star is being noticeably pulled by aninvisible partner that is more than 3 times as massive as the sun. a) If a redgiant (a dying star) is gravitationally accelerated at 0.075 m/s2 toward an object that is 9.4 � 1010 m away, how large a mass must the unseen bodypossess? b) How many times more massive is the object than the sun?(Ms � 1.99 � 1030 kg)

Answer: a.

Answer: b.

Exercise 10: The planet Saturn has a mass that is 95 times Earth’s mass and a radius that is9.4 times Earth’s radius. What is the acceleration due to gravity on Saturn?

Answer:


7-3 Escape SpeedVocabulary Escape Speed: The minimum speed an object must possess in order to escape

from the gravitational pull of a body.

In Chapter 5, you worked with gravitational potential energy and kineticenergy. When an object moves away from Earth, its gravitational potentialenergy increases. Since its total energy is conserved, its kinetic energydecreases. When the object is close to Earth, the gravitational force on it is afairly constant mg. However, as you know, the gravitational force dropsrapidly as you get farther from Earth. If an object moves upward from Earthwith enough speed, it will never run out of kinetic energy and will escapefrom Earth.

The escape speed for an object leaving the surface of any celestial body ofmass M and radius d is

v �

Notice that the mass of the escapingobject does not affect the escape speed.

Solved Examples

Example 6: Earth has a mass of 5.98 � 1024 kg and a radius of 6.38 � 106 km. What is theescape speed of a rocket launched on Earth?

Given: M � 5.98 � 1024 kg Unknown: v � ?d � 6.38 � 106 m Original equation: v �G � 6.67 � 10�11 N �m2/kg2

Solve: v � �

� 11 200 m/s

Any rocket trying to escape Earth’s gravitational pull must be going at least 11 200 m/s before engine cut-off, in order to get away.

Example 7: Compare Example 6 with the escape speed of a rocket launched from themoon. The mass of the moon is 7.35 � 1022 kg and the radius is 1.74 � 106 m.

Given: M � 7.35 � 1022 kg Unknown: v � ?d � 1.74 � 106 m Original equation: v �G � 6.67 � 10�11 N �m2/kg2 B2GM

d

B216.67 � 10�11 N #m2>kg2 2 15.98 � 1024 kg 26.38 � 106 mB2GM

d

B2GMd

B2GMd


Solve: v � � � 2370 m/s

Notice that you can escape from the moon by traveling much more slowlythan you must travel to escape the gravitational pull of Earth. This is whylaunching a Lunar Module from the moon’s surface was so much easier thanlaunching an Apollo spacecraft from Earth.

Practice Exercise

Exercise 11: How fast would you need to travel a) to escape the gravitational pull of thesun? (MS � 1.99 � 1030 kg, dS � 6.96 � 108 m) b) As the sun begins to die, itwill become a red giant. This means that its mass will remain the same but itsdiameter will increase substantially (perhaps even out as far as Earth’s orbit!).When the sun becomes a red giant, will its escape speed be greater than, lessthan, or the same as, it is now?

Answer: a.

Answer: b.

Exercise 12: How fast would the moon need to travel in order to escape the gravitationalpull of Earth, if Earth has a mass of 5.98 � 1024 kg and the distance fromEarth to the moon is 3.84 � 108 m?

Answer:

B216.67 � 10�11 N #m2>kg2 2 17.35 � 1022 kg 21.74 � 106 mB2GM

d


2216.67 � 10�11 N #m2>kg2 2 11.99 � 1030 kg 2 > 16.96 � 108 m 222GM>d

2216.67 � 10�11 N #m2>kg2 2 15.98 � 1024 kg 2 > 13.84 � 108 m 222GM>d

Exercise 13: What is the escape speed needed a) to escape the gravitational pull ofAsteroid B612 (see Exercise 7)? b) What would happen if you jumped up onAsteroid B612?

Answer: a.

Answer: b.

Exercise 14: Scotty finds it difficult to play catch on planet Apgar because the planet’sescape speed is only 5.00 m/s, and if Scotty throws the ball too hard, it fliesaway. If planet Apgar has a mass of 1.56 � 1015 kg, what is its radius?

Answer:


A-1: Halley’s Comet orbits the sun about every 75 years due to the gravitationalforce the sun provides. Compare the gravitational force between Halley’sComet and the sun when the comet is at aphelion (its greatest distance fromthe sun) and d is about 4.5 � 1012 m to the force at perihelion (or closestapproach), where d is about 5.0 � 1010 m.

A-2: In Exercise A-1, what is the comet’s acceleration a) at aphelion? b) atperihelion? (MS � 1.99 � 1030 kg)


2216.67 � 10�11 N #m2>kg2 2 11.00 � 104 kg 2 > 120.0 m 222GM>d

A-3: An early planetary model of the hydrogen atom consisted of a 1.67 � 10�27-kg proton in the nucleus and a 9.11 � 10�31-kgelectron in orbit around it at a distance of 5.0 � 10�11 m. Inthis model, what is the gravitational force between a protonand an electron?

A-4: At what height above Earth would a 400.0-kg weather satellite have to orbit inorder to experience a gravitational force half as strong as that on the surface ofEarth?

A-5: It is said that people often behave in unusual ways during a full moon. a) Calculate the gravitational force that the moon would exert on a 50.0-kgstudent in your physics class. The moon is 3.84 � 108 m from Earth and has amass of 7.35 � 1022 kg. b) Does the moon attract the student with a force thatis greater than, less than, or the same as the force with which the studentattracts the moon?

A-6: The tiny planet Mercury has a radius of 2400 km and a mass of 3.3 � 1023 kg.a) What would be the gravitational acceleration of an astronaut standing onthe surface of Mercury? b) Compare the motion of a ball dropped on thesurface of Mercury to that of a ball dropped on Earth.

A-7: The acceleration due to gravity on Venus is 0.89 that of Earth. a) If the radiusof Venus is 6.05 � 106 m, what is Venus’ mass? b) How does this compare toEarth’s mass? c) If you were on a diet and had to “weigh in,” would yourather stand on a scale on Venus or on Earth in order to appear as if you hadlost the most weight?

A-8: The planet Mars has a mass that is 0.11 times Earth’s mass and a radius that is0.54 times Earth’s radius. a) How much would a 60.0-kg astronaut weigh ifshe were to stand on the surface of Mars? b) Although Mercury is muchsmaller than Mars, it has almost the same gravitational acceleration. Describehow you might explain this phenomenon.

A-9: On October 26, 2000, the NEAR Shoemaker spacecraft swooped within 3 miles of the asteroid Eros, taking images and collecting data from a distancecloser than any spacecraft has ever come to an asteroid. Eros has a mass of6.69 � 1015 kg. The strange potato-like shape of Eros makes its diameterdifficult to determine. If the NEAR spacecraft is orbiting a distance of 18 300 m from Eros’ center of mass, what gravitational acceleration does Eros provide on NEAR?

A-10: Find the NEAR spacecraft’s escape speed from Eros, using the informationgiven in A-9.

A-11: NASA has announced that a mission to Mars to return rock samples to Earthcould come as early as 2011. If NASA landed a 360.-kg spacecraft on thesurface of Mars a) what would be the weight of the spacecraft on the planet’ssurface b) what escape speed would be needed for the craft to leave the planet and head back to Earth with its rock samples. (Mm � 6.42 � 1023 kg,dM � 3.39 � 106 m)



B-1: At what distance from Earth’s center must a spacecraft be in order toexperience the same gravitational attraction from both Earth and the moonwhen directly between the two? (ME � 5.98 � 1024 kg, MM � 7.35 �1022 kg dE�M � 3.84 � 108 m)

B-2: Jupiter’s innermost Galilean satellite, Io, is covered with active volcanoes,which exist because of the immense gravitational tugging on the satellite byJupiter and the other moons near Io. Io orbits 4.2 � 108 m from the center ofJupiter. The other Galilean satellites are located as follows from Jupiter’s center.Europa: 6.7 � 108 m, Ganymede: 1.0 � 109 m, and Callisto: 1.9 � 109 m. IfJupiter and its satellites are lined up as shown, what gravitational force doesthe satellite Io experience? (MI � 8.9 � 1022 kg, ME � 4.9 � 1022 kg,MG � 1.5 � 1024 kg, MC � 1.1 � 1023 kg, MJ � 1.9 � 1027 kg)

B-3: Saturn’s satellite, Titan, orbits the planet in a little less than 16 days. Titanorbits Saturn at an average distance of 1.216 � 109 m from the center of theplanet. Use this information to find the mass of Saturn.


8 Special Relativity

109

8-1 Time Dilation

Relative Speed

The speed of an object depends upon what frame of reference you use tomeasure that speed. If a ball is thrown forward out of a stationary car at 4 m/s,the ball will continue to travel horizontally at 4 m/s until it hits the ground.However, if the car is in motion, a number of different things can happen.

For example, according to a stationary observer, if the car is moving forwardat 10 m/s, the ball is also traveling 10 m/s in addition to the 4 m/s given tothe ball when it is thrown. Therefore, it has a speed relative to the observer of14 m/s.

Now, if the ball is thrown at 4 m/s in a direction opposite to the car’s motion,the initial 4 m/s with which the ball is thrown is subtracted from the 10 m/sspeed of the forward-moving car. The ball has a speed of 6 m/s relative to astationary observer. It is still moving in the same direction as the car but at areduced speed with respect to the ground.

Time Dilation

When an object (such as a spaceship) is traveling near the speed of light, thetime interval between two events that occur at the same place on the movingobject seems longer from the perspective of a stationary observer than it doesfrom the perspective of the moving observer. In other words, time appears tobe dilated, or stretched out. The stationary observer thinks that the traveler’sclock has slowed down. This dilation is written as

�t �

where �t is the time interval between two events, as measured by anobserver who is in motion with respect to the events;�to is the time interval between two events, as measured by anobserver who is at rest with respect to the events (also called theproper time);v is the speed of the moving object;c is the speed of light.

¢to21 � 1v2>c 2 2

Therefore, when a spaceship is traveling close to the speed of light, itsinhabitants will appear to age more slowly and, in fact, all events will occurmore slowly from the perspective of an Earth-based observer.

In physics, astronomical distances are often written with the unit light-years,(ly). A light-year is the distance light travels in 1 year. It is equivalent to9.46 � 1015 m.

Solved Examples

Example 1: Farmer MacGregor is throwing bales of hay off the back of his hay wagonwith a speed of 3 m/s relative to the wagon, which is pulled by a tractormoving forward with a speed of 7 m/s. With what horizontal velocity do thebales of hay hit the ground?

Solution: First, consider the direction of each of the velocities and treat themas vectors. Relative to the truck, the bales of hay are traveling at 3 m/s.However, relative to the ground, the speed is somewhat different, as shown.

v � 7 m/s � 3 m/s � 4 m/s (in the direction of the tractor’s motion)

Example 2: Monty is being pulled in his wagon with a speed of 2 m/s when he tosses infront of the wagon a Frisbee whose speed is 5 m/s relative to the ground.Neglecting air resistance, how fast is the Frisbee moving when his dog,Snoopy, catches it in his mouth?

Solution: The wagon is moving at 2 m/s while the Frisbee travels anadditional 5 m/s in the same direction. Therefore,

v � 2 m/s � 5 m/s � 7 m/s forward, relative to the ground.


Example 3: A light beam takes 3.0 � 10�8 s to bounce back and forth vertically between twomirrors inside a moving spaceship, accordingto an observer on board the spaceship. Howlong would the beam take according toGerard, a stationary observer on Earth, if thespacecraft were moving directly overhead ina direction perpendicular to the line of sightwith a speed of 0.60c?

Solution: The number 0.60c in the exercisemeans that the speed of the spacecraft is6/10 the speed of light. The speed of light isrepresented with the letter c.

Given: �to � 3.0 � 10�8 s Unknown: �t � ?v � 0.60c Original equation: �t �

Solve: �t � � � �

� 4.7 � 10�8 s

Therefore, if the spacecraft is traveling at 0.60c, a time interval of 3.0 � 10�8 saccording to the clocks on the spacecraft actually takes 4.7 � 10�8 s accordingto the clocks on Earth.

Practice Exercises

Exercise 1: Fiona is on her way home from France but she must leave her new-found love, Pierre,behind. As Fiona’s train pulls out of thestation at 4 m/s, Pierre tosses Fiona abouquet of flowers with a speed of 6 m/s.According to Fiona, how fast are the flowersmoving when she catches them?

Answer:

3.0 � 10�8 s20.64

3.0 � 10�8 s21 � 0.36

3.0 � 10�8 s21 � 3 10.60c 22>c2 4�to21 � 1v2>c 2 2

¢to21 � 1v2>c2 2

Special Relativity 111

Exercise 2: Skip is bringing his boat into port with a speed of 7 m/s and as he nears thedock, he tosses a tow rope from the bow with a speed of 3 m/s to a waitingdock worker. How fast is the rope moving when it is caught by the dockworker?

Answer:

Exercise 3: Superman leaves Lois in Metropolis to rescue a malfunctioning space probesent up from Earth. Flying at a speed of 0.70c, Superman reaches the probe in20. hours according to his wristwatch. How long would the trip takeaccording to Lois’s clock on Earth?

Answer:

Exercise 4: An elementary particle called a pion has a lifetime of 2.6 � 10�8 s when atrest. a) Will its lifetime be longer or shorter, as viewed from the stationaryframe of reference, if it is made to travel at 0.80c? b) What will its lifetime beaccording to a stationary observer?

Answer: a.

Answer: b.


21 � 3 10.70c 22>c2 421 � 1v2>c2 2

21 � 3 10.80c 22>c2 421 � 1v2>c2 2

Exercise 5: It is the year 3539 and, on his 30th birthday, Albert leaves for a new jobopening on the planet Zil. After saying good-bye to his twin brother Henry,Albert jumps in his spacecraft and takes off for Zil traveling at a speed of0.95c. The total trip takes 3.0 years according to the clocks on board Albert’sspaceship. How old are Henry and Albert when the three-year journey iscomplete?

Answer:

Exercise 6: The brightest star visible from the northern hemisphere is the star Sirius,which is 8.7 light-years from Earth in the constellation of Canis Major. It takesa spaceship 4.9 y to travel from Earth to Sirius, according to the spaceship’son-board clocks. According to Earth clocks, the trip takes 10.0 years. At whatfraction of the speed of light did the spacecraft travel?

Answer:

8-2 Relativistic Length and Energy

Length Contraction

When an object is traveling close to the speed of light, the length of that objectalong the direction of motion appears to shrink as seen by a stationaryobserver. In other words, length appears to contract. This contraction iswritten as

L � Lo 21 � 1v2>c2 2Special Relativity 113

21 � 3 10.95c 22>c2 421 � 1v2>c2 2

21 � 3 14.9 y 22> 110.0 y 22 421 � 1�to2>�t2 2

where L is the length (or distance) between two points as measured by anobserver who is in motion with respect to the points;Lo is the length (or distance) between two points as measured by anobserver who is at rest with respect to the two points (also called theproper length);v is the speed of the moving object;c is the speed of light.

Therefore, when a spaceship is traveling close to the speed of light, thespaceship itself will seem shorter, from the perspective of an Earth-basedobserver.

Energy

Einstein proposed the idea that anything that has mass has energy. Thisenergy is called the rest energy, Eo. It is measured in joules.

rest energy � (rest mass)(speed of light)2 or Eo � moc2

If the object is moving at a speed v, the total energy of the object is greater, asmeasured by a stationary observer.

E � mc2 � �

where E is the total energy of an object as measured by an observer who isin motion with respect to the object;Eo is the total energy of an object as measured by an observer who isat rest with respect to the object;m is the mass of the object as measured by an observer who is inmotion with respect to the object;mo is the mass of the object as measured by an observer who is atrest with respect to the object;v is the speed of the moving object;c is the speed of light.

moc221 � 1v2>c 2 2Eo21 � 1v2>c 2 2


Solved Examples

Example 4: The star Betelgeuse in the constellation of Orion is 520 light-years away asperceived by an observer on Earth. If a space traveler journeyed to Betelgeuseat 99% the speed of light (0.99c), how long would this distance be according tothe traveler?

Given: Lo � 520 ly Unknown: L � ?v � 0.99c Original equation: L � Lo

Solve: L � Lo � (520 ly) � (520 ly)� 73 ly

So the distance to the star would appear considerably shorter to the spacetraveler.

Example 5: An electron and a positron meet, each with a rest mass of 9.11 � 10�31 kg, andare converted to energy (gamma rays). How much energy is converted fromthe rest energy into gamma rays in the collision?

Solution: Because the mass in this exercise is actually the combination of theelectron’s mass and the positron’s mass, add these two masses together toobtain 1.82 � 10�30 kg.

Given: mo � 1.82 � 10�30 kg Unknown: E � ?c � 3.00 � 108 m/s Original equation: E � moc2

Solve: E � moc2 � (1.82 � 10�30 kg)(3.00 � 108 m/s)2 � 1.64 � 10�13 J

Example 6: A 10 000.-kg meteor falls to Earth from space. a) What is the rest energy of themeteor? b) When it is traveling at a speed of 0.0400c, what is the meteor’senergy according to an observer on Earth?

a. Given: mo � 10 000. kg Unknown: Eo � ?c � 3.00 � 108 m/s Original equation: Eo � moc2

Solve: Eo � moc2 � (10 000. kg)(3.00 � 108 m/s)2 � 9.00 � 1021 J

b. Given: Eo � 9.00 � 1021 J Unknown: E � ?v � 0.0400c Original equation: E �

Solve: E � � � 9.01 � 1021 J9.00 � 1021 J21 � 3 10.0400c 22>c2 4

Eo21 � 1v2>c 2 2

Eo21 � 1v2>c 2 2

20.1421 � 3 10.99c 22>c2 421 � 1v2>c2 221 � 1v2>c2 2


Practice Exercises

Exercise 7: The year is 2100, and a sports car company claims to have invented a new carthat can travel at 0.500c. You take one of these cars for a test drive past yourhouse, which is 15.0 m wide. How wide does your house appear to be whenthe car is up to full speed?

Answer:

Exercise 8: A stretch limo of the future is 8.0 m long but appears to be only 6.0 m longwhen driven at speeds near the speed of light. How fast must Linda the limodriver be going to make the limo appear 6.0 m long to an outside observer?

Answer:

Exercise 9: The starship Enterprise is traveling past Jupiter at a speed of 0.7500c. a) IfJupiter has a diameter of 142 796 km, how wide is Jupiter according to thecrew of the Enterprise? b) What shape will Jupiter appear to have?

Answer: a.

Answer: b.

Exercise 10: The net result of a hydrogen fusion reaction is that four hydrogen atomscombine to form one helium atom. The mass lost when rest energy isconverted into radiation energy in the reaction is 4.59 � 10 �29 kg. a) Howmuch radiation energy does this reaction produce? b) In what form can thisenergy be observed here on Earth?

Answer: a.

Answer: b.


L � Lo � (15.0 m) � 13.0 m21 � 3 10.500c 22>c2 421 � 1v2>c2 2

21 � 3 16.0 m 22> 18.0 m 22 421 � 1L2>Lo2 2

21 � 3 10.7500c 22>c2 421 � 1v2>c2 2

Exercise 11: A nuclear reactor releases 9.1 � 1013 J of energy during fission. How muchmass is needed to create this amount of energy?

Answer:

Exercise 12: In particle accelerators such as the CERN accelerator in Geneva, Switzerland,particles are accelerated to speeds near that of light. If a proton of rest mass1.67 � 10�27 kg travels at a speed of 0.95c, what is the total energy of theproton according to a stationary observer?

Answer:


A-1: Juan gets on the school bus in the morning and, as the driver starts to pullaway, Juan’s mother runs toward the bus with Juan’s lunch bag in her hand.When the bus is traveling at a speed of 12 m/s, Mom tosses the lunch bag toJuan and he reaches out the open window to catch it. If the bag is moving 3 m/s according to Juan, with what speed did Mom throw the lunch bag?

A-2: Ming and Wong are playing a game of table tennis in the recreation car of atrain. Each boy hits the ball with a speed of 20 m/s. a) If the train is travelingat 30 m/s, describe the speed of the ball as seen by an observer standing onthe ground behind the railroad crossing guardrail. b) How would Ming andWong describe the ball’s speed?

A-3: The year is 2092 and the beings of the planet Quigg have captured an alienwhom they are transporting home to show to the Quiggians. The trip takes 5.0 years, traveling at a speed of 0.80c according to the alien’s on-board clock.How long will the trip take according to the inhabitants of planet Quigg?

A-4: SpaceTours, Inc. is booking passage on a ship that will travel through space ata speed of 0.70c. The journey will last 5.0 years according to a stationaryobserver on Earth. How long will Margie, one of the passengers, be goneaccording to the on-board clock?


21 � 3 10.95c 22>c2 421 � 1v2>c2 2

A-5: As the Rebel Forces fly by the Death Star at a speed of 0.980c, what is theapparent diameter of the Death Star if its actual diameter is 7000. m?

A-6: Every time Pinocchio tells a lie, his nose grows 1.0 cm. In the past few weeksPinocchio has told many lies and his nose is now 10.0 cm long. How fast mustPinocchio travel to make his nose appear to be 2.0 cm long to a stationaryobserver?

A-7: Assume that all of the radiation energy in the Big Bang was converted into therest energy of the matter that is now the known universe. If the universe has arest mass of 1051 kg, how much radiation energy will be released if theuniverse eventually undergoes a “Big Crunch”?

A-8: At the Bates Linear Accelerator in Middleton, Massachusetts, electrons areaccelerated to near-light speeds inside a giant underground tunnel. If a9.11 � 10�31-kg electron is traveling at 0.89c, what is its total energy?


B-1: You are riding in a station wagon on a two-lane highway at 30 m/s, and youpass a sedan going in the other direction at 25 m/s.a) Why does the sedan appear to be moving so much faster than you are?b) How fast does the sedan appear to be moving from your perspective?c) How fast does your station wagon appear to be moving relative to the

sedan?d) How would your answers to (b) and (c) change if the sedan were moving

in the same direction as your car?e) How fast does each car appear to be moving to an observer standing by

the side of the road?

B-2: Leon observes that his heart beats 60.0 times per minute from his own frameof reference.a) If Leon gets into a rocket on his 15th birthday and flies away from Earth

fast enough so that his heartbeat appears to occur half as frequently asobserved from Earth, how fast is the rocket traveling?

b) How old is Leon if he returns to Earth after Earth-based clocks said that hehad been gone for 12.0 years?


9 Solids, Liquids, and Gases

119

9-1 DensityVocabulary Density: A measure of how much mass occupies a given space.

density � or D �

The SI unit for density is the kilogram per cubic meter (kg/m3).

Density is a characteristic property of a material. The density of an object doesnot change if the object is broken into smaller pieces. Although each piecenow has less mass than the original object, it has less volume as well.Therefore, the density remains the same.

Think of density as describing how “compact” an object is. Remember, thedensity of a material can change with temperature because atoms andmolecules move faster when they are heated, and thus usually occupy morespace.

Solved Examples

Example 1: While doing dishes, Zvi drops his 3.00 � 10�3-kg platinum wedding bandinto the dishwater, displacing a volume of 1.40 � 10�7 m3 of water. What isthe density of the plantinum band?

Given: m � 3.00 � 10�3 kg Unknown: D � ?V � 1.40 � 10�7 m3 Original equation: D �

Solve: D � � � 2.14 � 104 kg/m3

Example 2: At a temperature of 4 °C, 5000. kg of water will fill a volume of 5.000 m3.What is the density of water at 4 °C?

Given: m � 5000. kg Unknown: D � ?V � 5.000 m3 Original equation: D �

Solve: D � � � 1000. kg/m35000. kg

5.000 m3mV

mV

3.00 � 10�3 kg

1.40 � 10�7 m3mV

mV

mV

massvolume

Practice Exercises

Exercise 1: The planet Saturn has a mass of 5.69 � 1026 kg and a volume of 8.01 � 1023 m3.a) What is the density of Saturn? b) Would Saturn sink or float if you couldplace it in a gigantic bathtub filled with water?

Answer: a.

Answer: b.

Exercise 2: You are handed a 5.00 � 10�3-kg coin and told that it is gold. You discoverthat the coin has a volume of 5.90 � 10�7 m3. You know that the density ofgold is 19 300 kg/m3. Have you really been handed a gold coin, or simply agood imitation?

Answer:

Exercise 3: Diamond has a density of 3520 kg/m3. During a physics lab, a diamond drops out of Virginia’s necklace and falls into her graduated cylinder filledwith 5.00 � 10�5 m3 of water. This causes the water level to rise to the5.05 � 10�5�m3 mark. What is the mass of Virginia’s diamond?

Answer:

a) D � m/V � (5.69 � 1026 kg)/(8.01 � 1023 m3) � 710. kg/m3


Exercise 4: You are given three different liquids—water, oil and glycerin—and asked topredict which will occupy the top, middle, and bottom layers when all threeare poured into the same beaker. You take down the following data:

mass (in kg) volume (in m3)

water 0.1000 1.00 � 10�4

oil 0.0500 5.39 � 10�5

glycerin 0.0400 3.17 � 10�5

By finding the densities, determine how these liquids will layer themselves inthe beaker from top to bottom.

Answer:

9-2 Solids

Compression and Stretching

Vocabulary Elasticity: A property of a body that causes it to deform when a force isexerted and return to its original shape when the deforming force is removed,within certain limits.

Vocabulary Stress: The force exerted on an area divided by the area.

stress � �

The SI unit of stress is the newton per square meter (N/m2).

Vocabulary Strain: The ratio of change in dimension to original dimension.

Most often strain is used in describing the change in length of an object whena force is exerted.

strain � �¢LL

change in length

original length

FA

forcearea

Solids, Liquids, and Gases 121

Because strain is a ratio of lengths, it has no units.

Stress and strain are proportional to each other, and their ratio is equal to theelasticity of the material. The elasticity of a material is called the stretchmodulus or Young’s modulus, Y.

Young’s modulus � or Y � �

The SI unit for Young’s modulus is the newton per square meter (N/m2).

Shearing

Shearing is another way of applying stress to an object to cause a distortion.However, this type of distortion is not one of dimension, but one of shape. Forexample, the book in Figure A will look like the one in Figure B when it issheared.

In this case, the shearing stress is the force exerted on the area of one of thepages and the strain is the ratio of the horizontal distance the book moves, �L,to the original width of the book, L. The ratio of stress to strain is equal to theelastic modulus or the shearing modulus, S.

shearing modulus � or S � �

Solved Examples

Example 3: Jason, the piano tuner, is tuning a 0.50-m-long steel piano wire of cross-sectional area 0.18 cm2 by stretching it with a force of 1200 N. By how muchdoes this lengthen the wire? (Ysteel � 2.0 � 1011 N/m2)

Solution: First, convert cm2 to m2. 0.18 cm2 � 1.8 � 10�5 m2

Given: L � 0.50 m Unknown: �L � ?F � 1200 N Original equation: Y �A � 1.8 � 10�5 m2

Y � 2.0 � 1011 N/m2

FLA¢L

FLA¢L

F>A¢L>L

shearing stressshearing strain

FLA¢L

F>A¢L>L

stressstrain


Solve: �L � � � 1.7 � 10�4 m

Example 4: While writing his history research paper, Brent reaches across the library table for his dictionary and pulls it toward himself by the edge of the topcover with a force of 16 N, displacing the cover by 0.02 m. The top of the 0.05-m-thick dictionary measures 0.05 m2. What is the shear modulus of the dictionary?

Given: F � 16 N Unknown: S � ?A � 0.05 m2 Original equation: S �L � 0.05 m

�L � 0.02 m

Solve: S � � � 800 N/m2

Practice Exercises

Exercise 5: Two leopards are fighting over a piece of meat they caught while hunting. The leopards pull on the meat muscle with a force of 100. N, stretching the0.10-m-long tendon by 0.0080 m. If the cross-sectional area of the tendon is1.0 � 10�5 m2, what is its stretch modulus?

Answer:

Exercise 6: Before heading out on her big date, Ling stands in front of the bathroommirror brushing her 0.25-m-long hair with a force of 2.0 N. If the cross-sectional area of a piece of hair is 1.0 � 10�7 m2, by how much does the hairstretch when it is brushed? (Yhair � 2.0 � 109 N)

Answer:

116 N 2 10.05 m 210.05 m2 2 10.02 m 2

FLA¢L

FLA¢L

11200 N 2 10.50 m 212.0 � 1011 N>m2 2 11.8 � 10�5 m2 2

FLYA


Exercise 7: When a piece of wood is distorted by a karate chop, the top of the board iscompressed while the bottom is stretched as shown. Therefore, you must firstconsider the change in length of the bottom of the board where the breakbegins. Chantal is a black belt in karate and she breaks a 30.0-cm piece ofwood with a force of 70.0 N, changing it in length by 4.0 � 10�4 cm. What isthe cross-sectional area of the piece of wood? (Ywood � 1.0 � 109 N/m2)

Answer:

Exercise 8: While Miss Levesque is erasing the blackboard with her 9.0 � 10�3-m2 eraser,the eraser is subjected to a great amount of shearing force. If a 2.0-cm-thickeraser is pushed with a horizontal force of 1.5 N, displacing the top of theeraser by 5.0 mm, what is the shear modulus of the eraser?

Answer:

Exercise 9: Jorge is running down the newly-waxed school hallway and tries to slideacross the floor in his sneakers. The 2.0-cm thick rubber soles each have across-sectional area of 0.020 m2 and are sheared with a force of 15 000 N. a) How much are the shoes displaced horizontally? b) Why does Jorge fallforward? (Srubber � 5.0 � 109 N/m2)

Answer: a.

Answer: b.

9-3 LiquidsVocabulary Hydrostatic Pressure: The pressure exerted on an object by a column of fluid.

The hydrostatic pressure depends upon the original atmospheric pressurepushing on the surface of the fluid, and upon the fluid’s density and height.


The farther an object is located below the surface of the fluid, the greater thepressure acting on it.

hydrostatic pressure �atmospheric pressure � (density)(acceleration due to gravity)(height)

Ph � Pa � Dgh

For these exercises, assume that normal atmospheric pressure is 1.01 � 105 Pa.

Recall from Chapter 3 that a pascal (Pa) is equivalent to a newton per squaremeter (N/m2).

Archimedes’ Principle

According to Archimedes’ principle, an object completely or partiallyimmersed in a fluid is pushed up by a force that is equal to the weight of thedisplaced fluid.

buoyant force � (density)(acceleration due to gravity)(volume)

Fb � DgV

Here the density and volume are those of the displaced fluid. This equationcan be used whether the object sinks or floats. However, if the object is onlypartially submerged, the volume used in the calculation is that of thesubmerged portion. Therefore, for a floating object, the buoyant force is equalto the weight of the object itself.

Pascal’s Principle

According to Pascal’s principle, the change inpressure on one part of a confined fluid isequal to the change in pressure on any otherpart of the confined fluid.

�P �

Therefore, a small force exerted over a smallarea will result in a large force exerted over alarge area. An application of Pascal’s principleis found in hydraulic lifts, which are used to raise automobiles off the ground.In a hydraulic lift, the force exerted on a smaller piston provides a pressurethat is applied, undiminished, to the larger piston, enabling it to lift the car.

F1

A1�

F2

A2


Solved Examples

Example 5: Wanda watches the fish in her fish tank and notices that the angel fish like tofeed at the water’s surface, while the catfish feed 0.300 m below at the bottomof the tank. If the average density of the water in the tank is 1000. kg/m3,what is the pressure on the catfish?

Solution: Solve this exercise using the equation for hydrostatic pressure.

Given: Pa � 1.01 � 105 Pa Unknown: Ph � ?D � 1000. kg/m3 Original equation: Ph � Pa � Dghg � 10.0 m/s2

h � 0.300 m

Solve: Ph � Pa � Dgh � (1.01 � 105 Pa) � (1000. kg/m3)(10.0 m/s2)(0.300 m)� 1.01 � 105 Pa � 3.00 � 103 Pa � 1.04 � 105 Pa

Example 6: Phyllis is being fed intravenously in her hospital bed from a bottle 0.400 mabove her arm that contains a nutrient solution whose density is 1025 kg/m3.What is the pressure of the fluid that is going into Phyllis’ arm?

Given: D � 1025 kg/m3 Unknown: Pa � ?h � 0.400 m Original equation: Ph � Pa � Dghg � 10.0 m/s2

Pa � 1.01 � 105 Pa

Solve: Ph � Pa � Dgh � (1.01 � 105 Pa) � (1025 kg/m3)(10.0 m/s2)(0.400 m)� 1.01 � 105 Pa � 4.10 � 103 Pa � 1.05 � 105 Pa

Example 7: Palmer drops an ice cube into his glass of water. The ice, whose density is 917 kg/m3, has dimensions of 0.030 m � 0.020 m � 0.020 m, as shown in thediagram. What is the buoyant force acting on the ice?

Solution: Solve this exercise using Archimedes’principle.

The dimensions of the ice provide you with theice cube’s volume.

0.030 m � 0.020 m � 0.020 m � 1.2 � 10�5 m3

Because the density of the water is less than the ice, the ice will float so thatpart of it is above the surface. Therefore,

buoyant force � weight of water displaced � weight of ice


Given: Dice � 917 kg/m3 Unknown: Fb � ?g � 10.0 m/2 Original equation: Fb � DgVV � 1.2 � 10 �5 m3

Solve: Fb � DicegV � (917 kg/m3)(10.0 m/s2)(1.2 � 10�5 m3) � 0.11 N

Therefore, the water pushes the ice cube up with a force of 0.11 N.

Example 8: Every Sunday morning, Dad takes the family trash to the trash compactor inthe basement. When he presses the button on the front of the compactor, aforce of 350 N pushes down on the 1.3-cm2 input piston, causing a force of 22 076 N to crush the trash. What is the area of the output piston that crushesthe trash?

Solution: Solve this exercise using Pascal’s principle.

Given: F1 � 350 N Unknown: A2 � ?A1 � 1.3 cm2 Original equation: �F2 � 22 076 N

Solve: A2 � � � 82 cm2

Practice Exercises

Exercise 10: The head of a giraffe is 2.0 m above its heart and the density of the blood is1.05 � 103 kg/m3. What is the difference in pressure between the giraffe’sheart and head? (Fortunately, a giraffe’s neck has a special circulatory systemto adapt to this neck length, producing an even flow of blood to the head.)

Answer:

Exercise 11: How much pressure is needed in ground-based water pipes to pump waterup to the restaurant on the top floor of the World Trade Center, 410 m abovethe ground?

Answer:

122 076 N 2 11.3 cm2 2350 N

F2A1

F1

F2

A2

F1

A1


Exercise 12: The difference in pressure between the atmosphere and the human lungs is1.05 � 105 Pa. What is the longest straw you could use to draw up milkwhose density is 1030 kg/m3?

Answer:

Exercise 13: Cadir is basting a roast turkey with a meat baster that creates a pressure of9.980 � 104 Pa when the plastic bulb is squeezed and released. If turkey juicerises 0.0900 m up the tube of the baster, what is the density of the juice?

Answer:

Exercise 14: A 5450-m3 blimp circles Fenway Park during the World Series, suspended inEarth’s 1.21-kg/m3 atmosphere. The density of the helium in the blimp is0.178 kg/m3. a) What is the buoyant force that suspends the blimp in the air?b) How does this buoyant force compare to the blimp’s weight? c) How muchweight, in addition to the helium, can the blimp carry and still continue tomaintain a constant altitude?

Answer: a.

Answer: b.

Answer: c.


Exercise 15: Ivory soap will float when placed in water so that most of the soap issuspended below the surface, and only a small fraction sticks up above thewater line. A bar of soap has dimensions of 9.00 cm � 6.00 cm � 3.00 cm, as shown, and a density of 994 kg/m3. What is the buoyant force acting onthe soap?

Answer:

Exercise 16: Eliza, the auto mechanic, is raising a 1200.-kg car on her hydraulic lift so thatshe can work underneath. If the area of the input piston is 12.0 cm2, while theoutput piston has an area of 700. cm2, what force must be exerted on the inputpiston to lift the car?

Answer:

Exercise 17: Allegra’s favorite ride at the Barrel-O-Fun Amusement Park is the FlyingUmbrella, which is lifted by a hydraulic jack. The operator activates the rideby applying a force of 72 N to a 3.0-cm-wide cylindrical piston, which holdsthe 20 000.-N ride off the ground. What is the diameter of the piston thatholds the ride?

Answer:


2120 000 N 2 10.015 m 22> 172 N 22F2r12>F1

9-4 GasesThe ideal gas law expresses the relationship between the pressure, volume,and temperature of a gas.

In the exercises in this chapter, the mass of the gas remains constant. You willbe examining relationships between changes in pressure, volume, ortemperature, using a combined form of the law that reads:

� or �

where the subscript “1” signifies the initial conditions and the subscript “2”signifies the final conditions.

When you do calculations with the ideal gas law, use the correct SI units.Temperature is measured in kelvins (K), pressure is measured in pascals (Pa),and volume is measured in cubic meters (m3). See Chapter 10 for anexplanation of the Kelvin temperature scale.

If the temperature remains constant, the relationship between changes inpressure and volume is known as Boyle’s law. Boyle’s law says that volumedecreases as the pressure increases. If the pressure remains constant, therelationship between changes in volume and temperature is known asCharles’ law. Charles’ law says that volume increases as the temperatureincreases.

Solved Examples

Example 9: To capture its prey, a whale will create a cylindrical wall of bubbles beneaththe surface of the water, trapping confused fish inside. If an air bubble has avolume of 5.0 cm3 at a depth where the water pressure is 2.00 � 105 Pa, whatis the volume of the bubble just before it breaks the surface of the water?

Solution: In this exercise, the temperature remains the same. Remove it fromboth sides of the equation.

Given: P1 � 2.00 � 105 Pa Unknown: V2 � ?V1 � 5.00 cm3 Original equation: P1V1 � P2V2P2 � 1.01 � 105 Pa

Solve: V2 � � � 9.90 cm3

Example 10: Tootie, a clown, carries a 2.00 � 10�3-m3 helium-filled mylar balloon from the295-K heated circus tent to the cold outdoors, where the temperature is 273 K.How much does the volume of the balloon decrease?

12.00 � 105 Pa 2 15.00 cm3 21.01 � 105 Pa

P1V1

P2

P2V2

T2

P1V1

T1

1Pressure2 2 1Volume2 2Temperature 2

1Pressure1 2 1Volume1 2Temperature 1


Solution: In this exercise the pressure remains constant. Therefore, remove itfrom both sides of the equation.

Given: V1 � 2.00 � 10�3 m3 Unknown: V2 � ?T1 � 295 K Original equation:T2 � 273 K

Solve: V2 � � � 1.85 � 10�3 m3

V1 � V2 � (2.00 � 10�3 m3) � (1.85 � 10�3 m3) � 0.15 � 10�3 m3

Example 11: Taylor is cooking a pot roast for dinner in a pressure cooker. Water willnormally boil at a temperature of 373 K and an atmospheric pressure of1.01 � 105 Pa. What is the boiling temperature inside the pot, when thepressure is increased to 1.28 � 105 Pa? The pot maintains a constant volume.

Solution: In this example the volume remains constant. Therefore, remove itfrom both sides of the equation.

Given: P1 � 1.01 � 105 Pa Unknown: T2 � ?T1 � 373 K Original equation:P2 � 1.28 � 105 Pa

Solve: T2 � � � 473 K

Practice Exercises

Exercise 18: The Caloric value of food is measured with a device called a bomb calorimeter.Oxygen is forced into this sealed container and kept at a constant volume. Oncethe internal pressure is increased to 1.50 � 105 Pa, a small piece of food insidethe calorimeter is ignited with a spark. As the food burns, the temperatureinside the sealed vessel rapidly increases from 293 K to 523 K. What is the newpressure of the gas inside the chamber when the temperature rises?

Answer:

11.28 � 105 Pa 2 1373 K 21.01 � 105 Pa

P2T1

P1

P1

T1�

P2

T2

12.00 � 10�3 m3 2 1273 K 2295 K

V1T2

T1

V1

T1�

V2

T2


Exercise 19: Brandon takes Yvonne on a surprise hot-air balloon ride for her birthday.However, once the pair is airborne, Yvonne announces that she is afraid ofheights. The 2200.-m3 balloon is filled to capacity with 350.0 K air at a heightwhere the surrounding air presure is 1.01 � 105 Pa. When Brandon turns offthe heating unit, the air in the balloon begins to cool and the balloondescends. a) Why do both the pressure and volume of the air in the balloonremain constant, even though the balloon’s air cools to a temperature of 300.0 K? b) Hot-air balloons are always made so that the bottom remains openthroughout the flight. By how much would the balloon’s volume change if theballoon could be manually closed as the temperature dropped to 300.0 K?(Assume atmospheric pressure remains constant.)

Answer: a.

Answer: b.

Exercise 20: During Annette’s first airplane ride, her plane ascends from sea level, wherecabin pressure is 1.01 � 105 Pa, to flying altitude, where the cabin pressuredrops slightly to 1.00 � 105 Pa despite pressurized conditions. Annette feels asensation in her middle ear, whose volume is 6.0 � 10�7 m3. a) What is thenew volume of air inside Annette’s middle ear? b) What could Annette do tocompensate for this change in volume?

Answer: a.

Answer: b.

Exercise 21: Theo has won a a new car on a game show, and when his shiny new vehiclearrives on a warm 301-K (28°C) fall day, the 0.016-m3 tires have an airpressure of 2.02 � 105 Pa. However, two weeks later, when the temperaturedrops to 273 K (0°C), Theo’s pressure gauge reads only 1.90 � 105 Pa. What isthe new volume of the car tires?

Answer:



A-1: A 1.9–kg piece of wood from a sunken pirate ship has a volume of2.16 � 10�3 m3. Will this piece of wood float to the surface of the water orremain submerged with the ship?

A-2: Ursula drops a 0.0330-kg ice cube into her glass of soda water. The ice cubehas dimensions of 3.0 cm � 3.0 cm � 4.0 cm. Does the ice cube float or sink inUrsula’s drink?

A-3: Which is more dense, a 20.0-g silver bullet that occupies a volume of 1.9 cm3,or the 5.98 � 1024-kg Earth, that occupies a volume of 1.08 � 1021 m3?

A-4: In her gymnastics routine, Regina dismounts from the uneven–parallel barsand lands straight-legged on the ground, compressing her 0.250–m–longfemur by 2.10 � 10�5 m. If the femur has a cross-sectional area of 3.00 � 10�4 m2 and the stress modulus of bone is 2.00 � 1010 N/m2, withhow much force does Regina hit the ground?

A-5: When they go swimming in their favorite waterhole, Jeb and Dixie like to swing over the wateron an old tire attached to a tree branch with a3.0-m nylon rope. If the diameter of the rope is2.00 cm, by how much does the rope stretchwhen 60.0-kg Dixie swings from it?(Ynylon � 3.7 � 109 N/m2)

A-6: Lucy is going skin diving to see coral off the coast of Mexico in sea water witha density of 1025 kg/m3. a) How great is the pressure pushing on Lucy at adepth of 20.0 m? b) How will the pressure change if Lucy swims deeper?

A-7: A water tower sits on the top of a hill and supplies water to the citizensbelow. The difference in pressure between the water tower and the Daileys’house is 1.1 � 105 Pa, while the difference in pressure between the tower andthe Stearns’ house is 3.2 � 105 Pa. a) Which house sits at a higher elevation,the Daileys’ or the Stearns’? b) What is the difference in elevation between thetwo houses?

A-8: Eileen is floating on her back in the beautiful blue Caribbean during herspring vacation. If Eileen’s density is 980 kg/m3 and she has a volume of0.060 m3, what is the buoyant force that supports her in the sea water ofdensity 1025 kg/m3?

A-9: While swimming in her backyard pool, Nicole attempts to hold a 0.9000-m3

inner tube completely submerged under the water. a) What buoyant force willbe exerted on the inner tube as Nicole attempts to force it under the water? b) When Nicole lets go of the inner tube, it pops up to the surface with a forceof 8990. N. What is the weight of the inner tube?


A-10: Irene is testing the strength of her model balsa wood bridge with a hydraulicpress before the National Contest in Denver. Irene exerts a force of 3.0 N on a1-cm-radius input piston, and a force is exerted on the 10.0-cm-radius outputpiston. If the bridge can withstand a force of 350 N before breaking, will thebridge survive the test and make it into the contest?

A-11: In exercise A-6, if Lucy were to foolishly hold her breath as she ascends to thewater’s surface, a) by how many times would the volume of her lungs change(assuming the water temperature remains constant)? b) Would her lungs becrushed or would they blow up like a balloon? c) What is the best way toascend after diving?

A-12: Dong-Jae is bottling his own root beer in his basement where the airtemperature is 315 K. The pressure inside each root beer bottle is 1.20 � 105 Pa,but the caps will pop off the bottles if the pressure inside exceeds 1.35 � 105 Pa.After the bottles are sealed and labeled, Dong-Jae stores them in his attic, whichheats up to 364 K on a hot summer day. What happens to the pressure insidethe bottles?


B-1: A 40.0-m-long steel elevator cable has a cross-sectional area of 4.0 � 10�4 m2

and is able to stretch 1.0 cm before breaking. If the elevator itself has a mass of1000. kg, how many 70.0-kg people can safely ride in the elevator?(Ysteel � 2.0 � 1011 N/m2)

B-2: A can of soda displaces 3.79 � 10�4 m3 of water when completely submerged.Each 0.018-kg can contains 3.54 � 10�4 m3 of soda. a) Compare the buoyantforce on a can of diet soda of density 1001 kg/m3 to the force on a can ofregular soda of density 1060. kg/m3. b) If many cans of diet and regular sodaare in a large tub of water and ice, how can you easily pick out the diet soda?

B-3: Saul ascends from the city of Tucson, Arizona, to the top of Kitt Peak, 2900 mabove sea level. Usually Saul will feel his ears “pop” as the pressure inside hisears attempts to maintain equilibrium with the surrounding air. However, onthis day Saul has a cold and his Eustachian tube is clogged, causing atremendous pressure behind his 4.0 � 10�5-m2 ear drum. a) What force doesSaul feel pushing on his ears? b) Is this pressure pushing in or out of his ear ashe ascends? (Dair � 1.20 kg/m3)

B-4: Hannah and her friends go fishing in her 1.20-m3 rowboat, which has a massof 100. kg. How many 60.0-kg people can get into the boat before the boatsinks?


10 Temperature and Heat

135

10-1 Temperature and ExpansionVocabulary Temperature: A quantity that you can measure with a thermometer.

There are many different scales for measuring the temperature of an object.The SI unit for temperature is the kelvin (K). The Kelvin scale is based onabsolute zero, a point at which the internal movement of an object’s atoms ormolecules is a minimum and no heat can be removed. An increase of onekelvin on the Kelvin scale is equal to an increase of one degree Celsius on theCelsius scale. Respectively, the freezing and boiling temperatures of water onthese two scales are 0°C � 273 K and 100°C � 373 K.

Notice that the Kelvin scale does not use the degree symbol, °. We say, “Zerodegrees Celsius equals two hundred seventy-three kelvins.”

However, the Fahrenheit scale is most common on household thermometers.Degrees Fahrenheit can be changed to degrees Celsius by writing.

TC � (TF � 32.0)

Degrees Celsius can be changed to degrees Fahrenheit by writing

TF � a TCb � 32.0

Heating an object will generally make its atoms or molecules move faster andcause the object to increase in size.

Linear Expansion

When a solid object experiences a temperature change, its length will increaseby a certain amount depending upon the nature of the material.

change in length �

(original length)(coefficient of expansion)(change in temperature)

or �L � Lo�T

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where �, the coefficient of linear expansion, is a characteristic property of thematerial. The SI unit for the coefficient of linear expansion is °C�1 (which isthe same as 1/°C).

Area Expansion

An object may also expand in area when heated. The equation for areaexpansion is

change in area �

2(original area)(coefficient of linear expansion)(change in temperature)

or �A � 2Ao��T

Volume Expansion

If the volume of a solid or liquid expands, the equation is written as

change in volume �

(original volume)(coefficient of volume expansion)(change in temperature)

or �V � Vo ��T

where � is the coefficient of volume expansion.

Solved Examples

Example 1: Justin is trying to convince his mother that he has a fever and should stayhome from school. However, he has a thermometer that will measure histemperature in degrees Celsius. If Justin’s temperature is 39.0°C and “normal”is 98.6°F, is Justin’s temperature high enough to keep him home?

Given: TC � 39.0°C Unknown: TF � ?Original equation: TF � a TCb � 32.0

Solve: TF � a TCb � 32.0 � (39.0°C) � 32.0 � 102°F

Yes. He should stay home.

Example 2: The layer of the sun that we see is called the photosphere. It has atemperature of 5600 K. What is the sun’s temperature a) in degrees Celsius? b) in degrees Fahrenheit?

a. Given: TK � 5600 K Unknown: TC � ?Original equation: TC � TK � 273

Solve: TC � TK � 273 � 5600 � 273 � 5327°C

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b. Given: TC � 5327°C Unknown: TF � ?Original equation: TF � a TCb � 32.0

Solve: TF � a TCb � 32.0 � (5327°C) � 32.0 � 9621�F Pretty hot!

Example 3: Ernesto is knitting his wife a sweater in his 18°C air-conditioned living roomwith 0.30-m-long aluminum knitting needles, when he decides to knit outsidein the 27°C air. How much will the knitting needles expand when Ernestotakes them outside? (aluminum � 24 � 10�6 °C�1)

Given: Lo � 0.30 m Unknown: �L � ?� � 24 � 10�6 °C�1 Original equation: �L � Lo�T

To � 18°CTf � 27°C

Solve: �L � Lo��T � Lo�(Tf � To) � (0.30 m)(24 � 10�6 °C�1)(27°C � 18°C)

� 6.5 � 10�5 m

Example 4: Jacques, the French chef, is kneading the dough for French bread in his 21°Ckitchen. He places the dough on a 0.40-m � 0.60-m aluminum cookie sheet. Ifthe oven temperature is 177°C, how much does the cookie sheet expand inarea while it is in the oven? (�aluminum � 24 � 10�6 °C�1)

Solution: Because the cookie sheet will expand in two directions, it isnecessary to use the equation for area expansion. The area of the cookie sheetis 0.40 m � 0.60 m � 0.24 m2.

Given: Ao � 0.24 m2 Unknown: �A � ?� � 24 � 10�6 °C�1 Original equation: �A � 2Ao��T

To � 21°CTf � 177°C

Solve: �A � 2Ao��T � 2(0.24 m2)(24 � 10�6 °C�1)(177°C � 21°C)

� 0.0018 m2

Example 5: A thermometer contains 0.50 cm3 of mercury at room temperature (21°C)when Pilar takes it into the physics lab for an experiment. By how much doesthe volume of mercury in the thermometer change after it sits in an 80.°Cbeaker of water? (�mercury � 18 � 10�5 °C�1)

Given: V � 0.50 cm3 Unknown: �V � ?�mercury � 18 � 10�5 °C�1 Original equation: �V � Vo��T

To � 21°CTf � 80.°C

Solve: �V � Vo��T � Vo�(Tf � To) � (0.50 cm3)(18 � 10�5 °C�1)(80.°C � 21°C)

� 0.0053 cm3

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Temperature and Heat 137

Practice Exercises

Exercise 1: On a summer day at the equator on Mars, the temperature never rises higherthan 50.0°C. Find this temperature in degrees Fahrenheit in order todetermine if this would be a comfortable temperature for a human visitingMars.

Answer:

Exercise 2: The highest temperature ever recorded on Earth was 136.4°F at Al’ Aziziyah,Libya, on September 13, 1922. The lowest temperature ever recorded was�128.6°F at Vostok, Antarctica, on July 22, 1983. Calculate both of thesetemperatures in degrees Celsius.

Answer:

Exercise 3: The barium-yttrium ceramic compound used to demonstratesuperconductivity will work only if supercooled to a temperature of 125 K.What is the equivalent temperature a) in °C? b) in °F?

Answer:


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Exercise 4: Most bridges contain interlocking grates that allow the bridge to expand andcontract with the change in temperature. The Golden Gate Bridge in SanFrancisco is about 1350 m long. a) The seasonal temperature variation in SanFrancisco ranges from about 0°C to 30.°C. How much will the bridge expandbetween these extremes? (�steel � 12 � 10�6 °C�1) b) Approximately howwide is this gap compared to the length of an automobile?

Answer: a.

Answer: b.

Exercise 5: Selena has a fire in the fireplace to warm her 20.°C apartment. She realizesthat she has left the iron poker in the fire. How hot is the fire if the 0.60-mpoker lengthens 0.30 cm? (�iron � 12 � 10�6 °C�1)

Answer:

Exercise 6: Leila is building an aluminum-roofed shed in her backyard to store hergarden tools. The flat roof will measure 2.0 m � 3.0 m in area during thecoldest winter months when the temperature is �10°C, but temperatures inLeila’s neighborhood can reach as high as 38°C in the summer. What is thearea of the roof that should stick out from the shed in the summer so that theroof just fits the structure during cold winter nights?(�aluminum � 24 � 10�6 °C�1)

Answer:


Exercise 7: Just before midnight, when the air temperature is 10.0°C, Karl stops and fillsthe 0.0600-m3 gas tank of his car. At noon the next day, when the temperaturehas risen to 32.0°C, Karl finds a puddle of gasoline beneath his car. a) What doyou think happened? b) How much gasoline spilled out of Karl’s car (assuming no change in the volume of the tank)? (�gasoline � 3.00 � 10�4 °C�1)

Answer: a.

Answer: b.

10-2 HeatVocabulary Heat: The transfer of energy between two objects that differ in temperature.

Vocabulary Specific heat: A measure of the amount of heat needed to raise thetemperature of 1 kg of a substance by 1°C.

The common unit for specific heat is the joule per kilogram degree celsius(J/kg°C).

The transfer of heat from an object depends upon the object’s mass, thespecific heat, and the difference in temperature between the object and itssurroundings.

change in heat � (mass)(specific heat)(change in temperature)

or �Q � mc�T

The SI unit for heat is the joule (J). This is the same unit used for mechanicalenergy in Chapter 5.

The heat lost by one object equals the heat gained by another object.

Heat lost � Heat gained or (mc�T)lost � (mc�T)gained

For each object in the system, an mc�T term is needed.

Water has a very high specific heat. It makes a good cooling agent because ittakes a long time for water to absorb enough heat to greatly increase its


temperature. In the following exercises, you will need to know the specificheat of water and of ice.

cwater � 4187 J/kg°C cice � 2090 J/kg°C

All other values for specific heat will be given in the exercises.

Heat of Fusion

Vocabulary Heat of Fusion: The quantity of heat needed per kilogram to melt a solid (orsolidify a liquid) at a constant temperature and atmospheric pressure.

The amount of heat needed to melt a solid is

change in heat � (mass)(heat of fusion) or �Q � mhf

The SI unit for the heat of fusion is the joule per kilogram (J/kg).

For water, which will be used most frequently in the exercises, the heat offusion is 3.35 � 105 J/kg. This means that 3.35 � 105 J of heat is required toturn 1 kg of ice into water. The same amount of heat is given off when 1 kg ofwater turns into ice.

Heat of Vaporization

Vocabulary Heat of Vaporization: The quantity of heat needed per kilogram to vaporize aliquid (or liquify a gas) at a constant temperature and atmospheric pressure.

The amount of heat needed to vaporize a liquid is

change in heat � (mass)(heat of vaporization) or �Q � mhv

The SI unit for the heat of vaporization is the joule per kilogram (J/kg).

For water, the heat of vaporization is 2.26 � 106 J/kg. This is more than sixtimes the heat of fusion for water.

Note that “steam” is not the same thing as water vapor. Water vapor is aninvisible gas that results when water boils or evaporates. Steam is what yousee when water vapor is cooled and condenses back into water droplets.


Solved Examples

Example 6: Hypothermia can occur if the body temperature drops to 35.0°C, althoughpeople have been known to survive much lower temperatures. On January 19,1985, 2-year-old Michael Trode was found in the snow near his Milwaukeehome with a body temperature of 16.0°C. If Michael’s mass was 10.0 kg, howmuch heat did his body lose, assuming his normal body temperature was37.0°C? (chuman body � 3470 J/kg°C)

Given: m � 10.0 kg Unknown: �Q � ?c � 3470 J/kg°C Original equation: �Q � mc�T

Tf � 16.0°CTo � 37.0°C

Solve: �Q � mc�T � mc(Tf � To) � (10.0 kg)(3470 J/kg°C)(16.0°C � 37.0°C)� �729 000 J

The negative answer implies that there was a heat loss. The encouraging (andamazing) end to this example is that Michael survived!

Example 7: Gwyn’s bowl is filled with 0.175 kg of 60.0°C soup (mostly water) that shestirs with a 20.0°C silver spoon of mass 0.0400 kg. The spoon slips out of herhand and slides into the soup. What equilibrium temperature will be reachedif the spoon is allowed to remain in the soup and no heat is lost to the outsideair? (cspoon � 240. J/kg°C) Assume that the temperature of the bowl does notchange.

Given: mwater � 0.175 kg Unknown: Tf � ?cwater � 4187 J/kg°C Original equation: Heat lost � Heat gainedTwater � 60.0°C

mspoon � 0.0400 kgcspoon � 240. J/kg°CTspoon � 20.0°C

Solve: mc�Twater � mc�Tspoon

(0.175 kg)(4187 J/kg°C)(60.0°C � Tf) � (0.0400 kg)(240. J/kg°C)(Tf � 20.0°C)43 963 J � (732.7 Tf)J/°C � (9.6 Tf)J/°C � 192 J

44 155 J � (742.3 Tf)J/°C

Tf � � 59.5°C

Therefore, the temperature of the spoon and soup both reach equilibrium at59.5°C, so the spoon has become much hotter but the soup has only cooled by0.5°C.

Example 8: An igloo is made of 224 blocks of ice at 0°C, eachwith a mass of 12.0 kg. How much heat must begained by the ice to melt the entire igloo?

44 155 J742.3 J>°C


Solution: The total mass of the ice is 224 (12.0 kg) � 2690 kg

Given: m � 2690 kg Unknown: �Q � ?hf � 3.35 � 105 J/kg Original equation: �Q � mhf

Solve: �Q � mhf � (2690 kg)(3.35 � 105 J/kg) � 9.01 � 108 J

Example 9: Gus is cooking soup in his hot pot and finds that he has added too muchwater. If Gus needs to boil off 0.200 kg of water in order for his soup to havethe correct consistency, how much additional heat must Gus add once thesoup is boiling?

Given: m � 0.200 kg Unknown: �Q � ?hv � 2.26 � 106 J/kg Original equation: �Q � mhv

Solve: �Q � mhv � (0.200 kg)(2.26 � 106 J/kg) � 4.52 � 106 J

Example 10: To cool her 0.200-kg cup of 75.0°C hot chocolate (mostly water), Heidi drops a0.0300-kg ice cube at 0°C into her insulated foam cup. What is thetemperature of the hot chocolate after all the ice is melted?

Solution: The relationship “Heat lost � Heat gained” can take on many formsdepending upon what is happening in the exercise. In this exercise, heat is lostfrom the hot chocolate (mc�Twater) and gained by the ice cube, first melting it(mhf) and then raising its temperature (mc�Twater).

Given: mice � 0.0300 kg Unknown: Tf � ?mwater � 0.200 kg Original equation: Heat lost � Heat gained

hf � 3.35 � 105 J/kgcwater � 4187 J/kg°CTwater � 75.0°C

Tice � 0°C

Solve: mc�Twater � mhf(ice) � mc�Twater � (0.200 kg)(4187 J/kg°C)(75.0°C � Tf)

� (0.0300 kg)(3.35 � 105 J/kg) � (0.0300 kg)(4187 J/kg°C)(Tf � 0°C)

� 62 805 J � (837.4 Tf)J/°C � 10 050 J � 125.6 Tf(J/°C)

� 52 755 J � (963.0 Tf)J/°C so Tf � � 54.8°C52 755 J

963.0 J>°C


Practice Exercises

Exercise 8: Peter is heating water on the stove to boil eggs for a picnic. How much heat isrequired to raise the temperature of his 10.0-kg vat of water from 20.0°C to100.0°C?

Answer:

Exercise 9: Nova, whose mass is 50.0 kg, stays out skiing for too long and her bodytemperature drops by 2.00°C. What is the amount of heat lost from Nova’sbody? (chuman body � 3470 J/kg°C)

Answer:

Exercise 10: Phoebe’s insulated foam cup is filled with 0.150 kg of the coffee (mostlywater) that is too hot to drink, so she adds 0.010 kg of milk at 5.0°C. If thecoffee has an initial temperature of 70.0°C and the specific heat of milk is 3800 J/kg°C, how hot is the coffee after the milk is added? (Assume that noheat leaks out through the cup.)

Answer:

Exercise 11: Emily is testing her baby’s bath water and finds that it is too cold, so she addssome hot water from a kettle on the stove. If Emily adds 2.00 kg of water at80.0°C to 20.0 kg of bath water at 27.0°C, what is the final temperature of thebath water?

Answer:


Exercise 12: Finishing his ginger ale, Ramesh stands at a party holding his insulated foamcup that has nothing in it but 0.100 kg of ice at 0°C. How much heat must begained by the ice in order for all of it to melt?

Answer:

Exercise 13: In Exercise 12, how much more heat must be gained to raise the temperatureof the melted ice to room temperature of 23.0°C?

Answer:

Exercise 14: Under the spreading chestnut tree the village blacksmith dunks a red-hothorseshoe into a large bucket of 22.0°C water. How much heat was lost by thehorseshoe in vaporizing 0.0100 kg of water?

Answer:

Exercise 15: While Laurie is boiling water to cook spaghetti, the phone rings, and all 1.5 kgof water boils away during her conversation. If the water was initially at 15°C,how much heat must have been gained for all of it to turn into water vapor?

Answer:


Exercise 16: By January, the 3.0 kg of water in the birdbath in Robyn’s backyard has frozento a temperature of �7.0°C. As the season changes, how much heat must beadded to the water to make it a comfortable 25°C for the birds?

Answer:


A-1: The hottest temperature on a planet was 864°F recorded on Venus by theSoviet Venera probe and the U.S. Pioneer probe. The coldest place in the solarsystem is Pluto where the temperature is estimated at �360.0°F. Calculateeach of these temperatures in degrees Celsius.

A-2: The temperature of background radiation left over from the Big Bang duringthe creation of the universe is 3 K. What is the temperature of the universe a) in °C? b) in °F?

A-3: As he rides the train to work on a �4.0°C winter day, Mr. Trump notices thathe can hear the click of the train going over spaces between the rails. Sixmonths later, on a 30.0-°C summer day, the rails are pushed tightly togetherand he hears no click. If the rails are 5.00 m long, how large a gap is leftbetween the rails on the cold winter day? (�steel � 12 � 10�6 °C�1)

A-4: Bradley, working in his 23°C kitchen, is cooking himself a crepe in an ironskillet that has a circular bottom with a diameter of 30.00 cm. How hot mustthe skillet be in order for Bradley to make a 710.3-cm2 crepe that just fills thebottom of the pan? (�iron � 12 � 10�6 °C�1)

A-5: A popular winter activity of many college students is “traying,” or slidingdown a snow-covered hill on a tray borrowed from the dining hall. If Joanneremoves a 0.35-m � 0.65-m aluminum tray from the 20.°C dining hall to gotraying in the brisk �8°C winter air, how much will the tray shrink whentaken outside? (�aluminum � 24 � 10�6 °C�1)


A-6: A 0.50-m3 brass treasure chest is pulled out of thecold 15°C ocean and onto the deck of a ship,where the air temperature is 40°C. How muchdoes the volume of the treasure chest expand?(�brass � 56 � 10�6 °C�1)

A-7: Leslie takes a full bottle of benzene from the 25.0°C chemistry lab into the10.0°C refrigerated storage locker. Later, Leslie enters the storage locker andfinds that 37.0 cm3 of benzene is missing from the bottle. What was theoriginal volume of benzene in the bottle? (�benzene � 1240 � 10�6 °C�1)

A-8: Sidney is home from school with a cold, so Mom has made him a bowl ofchicken soup (mostly water), which she ladles from a pot into a glass bowl. If0.600 kg of soup at 90.0°C is placed in a 0.200-kg bowl that is initially at20.0°C, what will be the temperature of the soup when the bowl and souphave reached equilibrium? (cglass � 840. J/kg°C)

A-9: In A-8 above, when the soup and bowl are at 80.0°C, a chilled dumpling witha mass of 0.100 kg and a temperature of 10.0°C is added. What will be thetemperature of the dumpling, soup, and bowl when the three have reachedequilibrium? (cdumpling � 110. J/kg°C)

A-10: Nils is emptying the dishwasher. He removes a 0.200-kg glass that has atemperature of 30.0°C. Into it he pours 0.100 kg of diet soda (mostly water),which comes out of the refrigerator with a temperature of 5.00°C. Assumingno external heat loss, what will be the final equilibrium temperature of theglass of diet soda? (cglass � 840. J/kg°C)

A-11: In Exercise A-10, Nils doesn’t feel that his drink is cold enough, so he throwsin an ice cube whose temperature is �3.0°C. What is the mass of the ice cubeif his drink (and glass) are now cooled to 1.0°C?

A-12: A puddle filled with 20. kg of water is completely frozen to �6.0°C in themiddle of the winter. How much heat must be absorbed by the puddle to meltthe ice and warm the water up to 20.°C during the spring thaw?

A-13: Before ironing his shirt for work, Nathaniel drops some water on his iron totest whether it is hot enough to iron his clothes. How much heat is needed tovaporize a 5.0 � 10�4-kg drop of 20.°C water?



B-1: Lawrence, a civil engineer, uses a steel tape measure to figure the dimensionsof the Emersons’ property. When the temperature is 37°C, he determines theproperty line to be 152.000 m long. However, the length of the property seemsto have changed when Lawrence returns on a 5.0°C winter day. a) Does theproperty appear to be longer in the warm weather or the cold? Explain whyyou think this is so. b) How long is the property when measured by his steeltape in the winter? (�steel � 12 � 10�6 °C�1)

B-2: One cool 5.0°C spring morning,Mason lays a brick sidewalk up tohis house, placing the 25.0-cm-longbricks end to end against eachother. However, Mason forgets toleave a space for expansion andwhen the temperature reaches 36.0°C, the bricks buckle. How high will thebricks rise? (�brick � 10.0 � 10�6 °C�1)

B-3: Phil is making a sandwich and he is having trouble getting the lid off the jarof mayonnaise. a) If the steel lid and the glass jar each have a diameter of 10. cm at a room temperature of 21°C, should Phil run the lid under water thatis 20.°C warmer or 20.°C cooler to remove the lid? b) When he completes thecorrect procedure to free the lid, what is the size of the space between the lidand the jar? (�aluminum � 24 � 10�6 °C�1 and �glass � 8.5 � 10�6 °C�1)

B-4: Pablo brings a 6000.-cm3 aluminum can filled all the way to the top withturpentine up from the 20.0°C basement and sets it outside where he ispainting. The noonday sun heats the turpentine and the aluminum containerto 45.0°C. Will the turpentine overflow the container? If so, how much willspill out? If not, how much more could be added to the empty space created?(�aluminum � 77 � 10�6 °C�1 and �turpentine � 900. � 10�6 °C�1)

B-5: In a physics experiment, a 0.100-kg aluminum calorimeter cup holding 0.200 kg of ice is removed from a freezer, where both ice and cup have beencooled to �5.00°C. Next, 0.0500 kg of steam at 100.°C is added to the ice in the cup. What will be the equilibrium temperature of the system after the icehas melted? (caluminum � 920. J/kg°C)


11 Simple Harmonic Motion

149

11-1 SpringsVocabulary Period: The time it takes for a vibrating object to repeat its motion.

Vocabulary Frequency: The number of vibrations made per unit time.

Period and frequency are the reciprocals of each other. In other words,

T � and f �

Since period is a measure of time, its SI unit is the second, while the unit forfrequency is the reciprocal of this, or 1/second. Another way of writing 1/s iswith the unit hertz (Hz).

You may recognize this as being similar to the explanation of period andfrequency in Chapter 6 on circular motion.

Hooke’s Law

Whenever a spring is stretched from its equilibrium position and released, itwill move back and forth on either side of the equilibrium position. The forcethat pulls it back and attempts to restore the spring to equilibrium is calledthe restoring force. Its magnitude can be written as

restoring force � (force constant)(displacement from equilibrium)or F � kx

This relationship is known as Hooke’s law. The force constant is a measure ofthe stiffness of the spring. The SI unit for the force constant is the newton permeter (N/m).

Keep in mind that this is the force required to restore the spring back to itsoriginal position. The force that acts to move the spring away from theequilibrium position is equal in magnitude to the restoring force, but oppositein direction.

Simple harmonic motion is motion that occurs when the restoring forceacting on an object is proportional to the object’s displacement from its restposition. Objects at the end of springs move in simple harmonic motion whenthey are displaced from their rest position and bounce up and down on thespring, or oscillate.

1T

1f

Period of a Mass on a Spring in Simple Harmonic Motion

The only two things that affect the period of an object hanging on a bouncingspring are the object’s mass and the force constant of the spring on which theobject is oscillating.

Period � 2� or T � 2�

To prove that this equation does indeed give the period in seconds, simplifythe units for by writing

� � � s

Solved Examples

Example 1: A hummingbird beats its wings up and down with a frequency of 80.0 Hz.What is the period of the hummingbird’s flaps?

Given: f � 80.0 Hz Unknown: T � ?Original equation: T �

Solve: T � � � 0.0125 s

Example 2: In anticipation of her first game, Alesia pulls back the handle of a pinballmachine a distance of 5.0 cm. The force constant of the spring in the handle is200 N/m. How much force must Alesia exert?

Solution: First, convert cm to m. 5.0 cm � 0.050 m

Given: k � 200 N/m Unknown: F � ?x � 0.050 m Original equation: F � kx

Solve: F � kx � (200 N/m)(0.050 m) � 10 N

Example 3: As Bianca stands on a bathroom scale, whose force constantis 220 N/m, the needle on the scale vibrates from side toside. a) If Bianca has a mass of 180 kg, what is the period ofvibration of the needle as it comes to rest? b) If Bianca goeson a diet, how will this change the period of vibration?

180.0 Hz

1f

1f

2s2Rkg

kg #m>s2

mB kg

N>m

2m>k

BmkB mass

force constant


a. Given: m � 180 kg Unknown: T � ?k � 220 N/m Original equation: T � 2�

Solve: T � 2� � 2� � 5.7 s

In other words, this is the amount of time for one complete oscillation.

b. Because the mass will be smaller, the period of vibration will be smaller. Inother words, it will take less time for the needle on the scale to bounce fromside to side as it comes to rest.

Practice Exercises

Exercise 1: Terry jumps up and down on a trampoline with a frequency of 1.5 Hz. Whatis the period of Terry’s jumping?

Answer:

Exercise 2: Gary Stewart of Reading, Ohio set a pogo stick record in 1990 by jumping 177 737 times. a) If the pogo stick he used had a force constant of 6000. N/mand was compressed 0.12 m on each jump, what force must Gary have exertedon the pogo stick upon each jump? b) What force would be exerted back upon Gary each time he went up?

Answer: a.

Answer: b.

B 180 kg

220 N>mBmk

Bmk

Simple Harmonic Motion 151

Exercise 3: At the post office, Cliff, a postal worker, places a 0.60-kg package on a scale,compressing the scale by 0.03 m. a) What is the force constant of the spring inthe postal scale? b) What happens to the force constant if Cliff weighs aheavier package?

Answer: a.

Answer: b.

Exercise 4: A jack-in-the-box lid will pop open when a crank is turned on the outside ofthe box. If Jack pushes against the inside of the box with a force of 3.00 Nwhen the lid is closed, and the spring is compressed 10.0 cm fromequilibrium, what is the force constant of the spring?

Answer:

Exercise 5: Sam, a butcher, puts 3.0 kg of chopped beef on the 1.0-kg pan of his scale,which has a spring whose force constant is 400. N/m. What is the period ofvibration of the pan as it comes to rest? b) If Sam adds more beef to the scale,what will this do to the period of vibration?

Answer: a.

Answer: b.


214.0 kg 2 > 1400. N>m 22m>k

Exercise 6: A toy bobs up and down over Campbell’s crib with a period of 1.0 s. The toyhangs from the end of a spring whose force constant is 20.0 N/m. What is themass of the toy?

Answer:

11-2 PendulumsThe period of a pendulum depends only upon the pendulum’s length (if theangle of swing is not too large). A long pendulum has a longer period than ashort pendulum. The relationship between period and length can be shownwith the following equation.

Period � 2� or T � 2�

It should be noted that this equation works only for a pendulum whose massis considerably larger than the mass of the string from which it swings. Tosimplify calculations, in the following exercises you will be working withpendulums swinging from strings of negligible mass.

Solved Examples

Example 4: A tall, thin tree sways back and forth in the breeze with a frequency of 2 Hz.What is the period of the tree?

Given: f � 2 Hz Unknown: T � ?Original equation: T �

Solve: T � � � 0.5 s1

2 Hz1f

1f

BLgB length

acceleration due to gravity


Example 5: World-reknowned hypnotist Paulbar the Greatswings his watch from a 20.0-cm chain in frontof a subject’s eyes. What is the period of swingof the watch?


Given: L � 0.20 m Unknown: T � ?g � 10.0 m/s2 Original equation: T � 2�

Solve: T � 2� � 2� � 0.89 s

Therefore, it takes 0.89 s for the watch to swing in one direction and backagain, through one full cycle.

Example 6: A spider swings in the breeze from a silk thread with a period of 0.6 s. Howlong is the spider’s strand of silk?

Solution: The answer is determined using the pendulum equation,but now it must be set up in terms of the unknown, L. First, squareall of the terms to get rid of the radical. The equation becomes

T2 � 4�2 . Then rearrange the equation as shown.

Given: T � 0.60 s Unknown: L � ?g � 10.0 m/s2 Original equation: T � 2�

Solve: L � � � 0.09 m

Practice Exercises

Exercise 7: A metronome is a device used by many musicians to get the desired rhythmfor a musical piece. If a metronome is clicking back and forth with afrequency of 0.5 Hz, what is the period of the metronome?

Answer:

110.0 m>s2 2 10.6 s 224p2

gT2

4p2

BLg

Lg

B 0.20 m10.0 m>s2BL

g

BLg


Exercise 8: Many amusement parks feature a ride in which a giant ship swings back andforth. If the period of the ship is 8.00 s, what is the frequency of the swingingship?

Answer:

Exercise 9: Tegan, a trapeze artist, swings from a 2.5-m-long trapeze, high above thethree-ring circus. a) What is Tegan’s period of swing? b) Would Tegan’s periodof swing change if she were more massive? If so, how?

Answer: a.

Answer: b.

Exercise 10: Danielle is pushing her twin Daniel on a swing that hangs from a tree branchby 2.0-m-long ropes. With what frequency will Danielle have to push Danielas he swings?

Answer:


212.5 m 2 > 110.0 m>s2 22L>g

212.0 m 2 > 110.0 m>s2 22L>g

Exercise 11: Marla, a maid, is standing on the Vanderbilt’s dining room table dusting thechandelier. While Marla is reaching up, she slips and grabs hold of thechandelier to catch her balance. When she lets go, the chandelier begins toswing with a period of 1.6 s. How long is the cable connecting the chandelierto the ceiling?

Answer:

Exercise 12: You have been commissioned by NASA to travel to Jupiter’s innermostGalilean satellite, Io, to learn more about this volcanic moon. As you boardthe spacecraft, you are handed a rock tied to a 10.0-cm string, and astopwatch, and are asked to derive an experiment that would allow you todetermine the acceleration due to gravity on Io. You must use both pieces ofequipment and nothing more. a) Describe how you would calculate Io’sgravitational acceleration. b) If the pendulum swings with a period of 1.48 s,what is the gravitational acceleration on Io?

Answer: a.

Answer: b.



A-1: Mr. Knote, a piano tuner, taps his 440-Hz tuning fork with a mallet. What isthe period of the vibrating tuning fork?

A-2: Denny jumps up and down on his bed, taking 0.75 s foreach jump. What is the frequency of Denny’s jumping?

A-3: Inside most ball-point pens is a small spring thatcompresses as the pen is pressed against the paper. If aforce of 0.1 N compresses the pen’s spring a distance of0.005 m, what is the force constant of the tiny spring?

A-4: Maureen is trying to predict the period of a mass hung on a spring. She has aspring of negligible mass and four weights to hang on the end. Maureencollects the following data as she observes the stretch of the spring:

force (N) displacement (m)

2.5 0.0505.0 0.1027.5 0.149

10.0 0.199

a) Plot a graph of force (on the y-axis) vs. displacement (on the x-axis). b) Findthe slope of the graph. What does this slope represent? c) Use the informationyou have obtained to find the period of the spring when a 3.0 kg mass is hungon the end.

A-5: Kim drives her empty dump truck over a berm (also called a speed bump) atthe construction site. The truck has a mass of 3000. kg and the force constantfor one of the truck’s springs is 100 000. N/m. (Remember, the truck has 4wheels.) a) What is the resulting period of the bouncing truck as it goes overthe bump? b) If Kim leaves the construction site with a load of dirt in hertruck, what will this do to the period of her dump truck as it crosses theberm?

A-6: A monkey swings from a jungle vine by his 0.30-m-long tail. a) What is the period of swing of the monkey? b) With what frequency does the monkey swing?

A-7: A wrecking ball used to demolish buildings swingsfrom a 10.0-m-long cable. What is the period of thewrecking ball as it swings?

A-8: A crow attempts to land on a small bird feeder, causing it to swing back andforth with a frequency of 0.350 Hz. How long is the wire from which thefeeder hangs?


A-9: The acceleration due to gravity on the moon is 1/6 that on Earth. a) If youwanted a pendulum clock to tell time on the moon the same as it does onEarth (i.e., have the same period), would you need to lengthen or shorten thependulum? b) If the pendulum was originally 24.0 cm long on Earth, howlong should it be on the moon?


B-1: Ezra, a 60.0-kg high school student, is sleeping on his waterbed when his 2.0-kg cat, Muffin, jumps onto his back, causing Ezra to sink 2.0 cm deeperinto the waterbed. a) If Muffin then jumps off Ezra from this new equilibriumposition, what will be the period of Ezra’s bobbing motion on the waterbed?b) Will this period slow down, speed up, or remain the same as the amplitudeof the bounces gets smaller and smaller? Explain your answer.

B-2: Andy (mass 80.0 kg), Randy (mass 60.0 kg), and twins Candy and Mandy(each with a mass of 70.0 kg) climb into a 1000.-kg car, causing each of thefour springs to compress 4.00 cm. Find the period of vibration of the car as itcomes to rest after the four get in.

B-3: Tanja talks long distance with her boyfriend every night from her dormitorypay phone, and her phone bills are getting rather high. She has decided thatshe must limit each of her calls to 10 minutes. Since Tanja doesn’t have awatch, she devises a unique way to time her calls. Tanja notices that the payphones each have a cord that is 0.800 m long. Therefore, as she talks on onephone, she can swing the receiver of the adjacent phone to time her call. Howmany complete swings will the nearby phone receiver make before Tanjamust hang up?

B-4: On a 0°C-winter day, a 10.000-m-long brass Foucault pendulum hanging inthe covered entrance to the science museum swings back and forth with therotation of Earth. The outdoor temperature variations range from 0°C in thewinter to 30.0°C in the summer. How does the period of the pendulumchange throughout the year? (�brass � 19 � 10�6°C �1)

B-5: Gillian buys a pendulum clock at a discount store and discovers when shegets it home that it loses 6.00 minutes each day. a) Should she lengthen orshorten the pendulum in order for it to keep accurate time? b) If thependulum has a period of 2.00 s, by how much must the length be changed sothat the clock keeps accurate time?


12 Waves and Sound

159

12-1 Wave MotionVocabulary Wave: A disturbance in a medium.

In this chapter you will be working with waves that are periodic or thatrepeat in a regular, rhythmic pattern.

wave speed � AwavelengthBAfrequencyB or v � �f

The SI unit for wave speed is the meter per second (m/s). The speed of soundin air increases with air temperature. For the following exercises, the speed ofsound will be written as 340.0 m/s. All electromagnetic radiation includingradio waves and light waves travel at the speed of light, 3.00 � 108 m/s.

The wavelength of a wave is the distance from one point on a wave to thenext identical point on the same wave, for example, from crest to crest, troughto trough, or condensation to condensation. The symbol for wavelength is theGreek letter “lambda,” �.

The SI unit for wavelength is the meter (m), which is the same unit used forlength in earlier chapters.

The SI unit for frequency is the hertz (Hz). When talking about the broadcastfrequency of a radio station, frequencies of FM radio stations are given inmegahertz, or MHz, and frequencies of AM radio stations are given inkilohertz, or kHz.

1 MHz � 1 � 106 Hz and 1 kHz � 1 � 103 Hz

Solved Examples

Example 1: Radio station WKLB in Boston broadcasts at a frequency of 99.5 MHz. What isthe wavelength of the radio waves emitted by WKLB?

Given: v � 3.00 � 108 m/s Unknown: � � ?f � 99.5 � 106 Hz Original equation: v � �f

Solve: � � � � 3.02 m

Therefore, the distance from one point on the wave to the next identical pointon the same wave is 3.02 m.

Example 2: In California, Clay is surfing on a wave that propels him toward the beachwith a speed of 5.0 m/s. The wave crests are each 20. m apart. a) What is thefrequency of the water wave? b) What is the period?

a. Given: v � 5.0 m/s Unknown: f � ?� � 20. m Original equation: v � �f

Solve: f � � � 0.25 Hz

b. Given: f � 0.25 Hz Unknown: T � ?Original equation: T �

Solve: T � � � 4.0 s

One crest comes along every 4.0 s.

Practice Exercises

Exercise 1: Harriet is told by her doctor that her heart rate is 70.0 beats per minute. IfHarriet’s average blood flow in the aorta during systole is 1.5 � 10�2 m/s,what is the wavelength of the waves of blood in Harriet’s aorta, created byher beating heart?

Answer:

Exercise 2: Dogs are able to hear much higher frequencies than humans are capable ofdetecting. For this reason, dog whistles that are inaudible to the human earcan be heard easily by a dog. If a dog whistle has a frequencyof 3.0 � 104 Hz, what is the wavelength of the sound emitted?

Answer:

10.25 Hz

1f

1f

5.0 m>s20. m

vl

3.00 � 108 m>s99.5 � 106 Hz

vf

160 Waves and Sound

Exercise 3: While flying to Tucson, Connie’s plane experiences turbulence that causes thecoffee in her cup to oscillate back and forth 4 times each second. If the wavesof coffee have a wavelength of 0.1 m, what is the speed of a wave movingthrough the coffee?

Answer:

Exercise 4: At a country music festival in New Hampshire, the Oak Ridge Boys areplaying at the end of a crowded 184-m field when Ronny Fairchild hits a noteon the keyboard that has a frequency of 400. Hz. a) How many fullwavelengths are there between the stage and the last row of the crowd? b) How much delay is there between the time a note is played and the time itis heard in the last row?

Answer: a.

Answer: b.

12-2 Doppler EffectVocabulary Doppler Effect: A change in the apparent frequency of sound due to the

motion of the source of the receiver.

You probably associate the Doppler effect with the change in pitch (frequency)of a loud car or siren just as it passes you. The pitch suddenly drops just asthe object moves by. Light can also be Doppler shifted but the Doppler shift oflight will not be discussed in this chapter.

The equation that describes this effect can be used whether the source isapproaching or receding from the observer. It also works if either the sourceor observer is at rest, or if there is a chase situation in which both are movingin the same direction.

Waves and Sound 161

perceived frequency � actual frequency

or f � fo

Here, fo refers to the actual frequency being emitted by an object, while f is thefrequency heard by the observer as the source approaches or recedes. If asource approaches, the perceived frequency will be higher than the actualfrequency. If a source recedes, the perceived frequency is lower than the actualfrequency.

In order for this equation to work properly, there is a standard convention towhich you must adhere whenever solving Doppler exercises.

vo is (�) if the observer moves toward the source.vo is (�) if the observer moves away from the source.vs is (�) if the source moves toward the observer.vs is (�) if the source moves away from the observer.

Remember, it is not necessary to always have both the observer and thesource in motion. Often one will be moving and the other will be at rest.

Solved Examples

Example 3: Sitting on the beach at Coney Island oneafternoon, Sunny finds herself beneath theflight path of the airplanes leaving KennedyAirport. What frequency will Sunny hear asa jet, whose engines emit sound at afrequency of 1000. Hz, flies toward her at aspeed of 100.0 m/s?

Solution: First draw a diagram of thesituation. Notice in the calculation belowthat Sunny is sitting at rest and the plane isapproaching. Therefore, the source ismoving toward the observer. The observerremains stationary.

Given: fo � 1000. Hz Unknown: f � ?vo � 0 m/s Original equation: f � fov � 340. 0 m/s

vs � 100.0 m/s

Solve: f � fo � 1000. Hz � 1417 Hz1340.0 m>s � 0 m>s 21340.0 m>s � 100.0 m>s 2

1v � vo 21v � vs 2

1v � vo 21v � vs 2

1v � vo 21v � vs 2

1speed of sound � speed of observer 21speed of sound � speed of source 2

162 Waves and Sound

Example 4: In the previous example, what frequency willSunny observe as the jet travels away from her atthe same speed?

Solution: Again, draw a diagram of the situation.This time, the source is moving away from theobserver, so the value for vs must be negative.

Given: fo � 1000. Hz Unknown: f � ?vo � 0 m/s Original equation: f � fov � 340.0 m/s

vs � �100.0 m/s

Solve: f � fo � 1000. Hz � 772.7 Hz

Example 5: A sparrow chases a crow with a speed of 4.0 m/s, while chirping at afrequency of 850.0 Hz. What frequency of sound does the crow hear as he fliesaway from the sparrow with a speed of 3.0 m/s?

Given: fo � 850.0 Hz Unknown: f � ?vo � �3.0 m/s Original equation: f � fov � 340.0 m/s

vs � 4.0 m/s

Solve: f � fo � 850.0 Hz � 852.5 Hz

Therefore, since the sparrow is approaching the crow, the crow hears afrequency that is higher than the original.

Practice Exercises

Example 5: One foggy morning, Kenny is driving his speed boat toward the Brant Pointlighthouse at a speed of 15.0 m/s as the fog horn blows with a frequency of180.0 Hz. What frequency does Kenny hear as he moves?

Answer:

1340.0 m>s � 3�3.0 m>s 4 21340.0 m>s � 4.0 m>s 2

1v � vo 21v � vs 2

1v � vo 21v � vs 2

1340.0 m>s � 0 m>s 21340.0 m>s � 3�100.0 m>s 4 2

1v � vo 21v � vs 2

v � vov � vs

Waves and Sound 163

Example 6: Dad is driving the family station wagon to Grandma’s house when he getstired and pulls over in a roadside rest stop to take a nap. Junior, who is sittingin the back seat, watches the trucks go by on the highway and notices thatthey make a different sound when they are coming toward him than they dowhen they are moving away. a) If a truck with a frequency of 85.0 Hz istraveling toward Junior with a speed of 27.0 m/s, what frequency does Juniorhear as the truck approaches? b) After the truck passes, what frequency doesJunior hear as the truck moves away?

Answer: a.

Answer: b.

Exercise 7: One way to tell if a mosquito is about to sting is to listen for the Doppler shiftas the mosquito is flying. The buzzing sound of a mosquito’s wings is emittedat a frequency of 1050 Hz. a) If you hear a frequency of 1034 Hz, does thismean that the mosquito is coming in for a landing or that it has just bitten youand is flying away? b) At what velocity is the mosquito flying?

Answer: a.

Answer: b.

164 Waves and Sound

Exercise 8: Barney, a bumblebee flying at 6.00 m/s, is being chased by Betsy, a bumblebeewho is flying at 4.00 m/s. Barney’s wings beat with a frequency of 90.0 Hz.What frequency does Betsy hear as she flies after Barney?

Answer:

Exercise 9: Mrs. Gonzalez is about to give birth and Mr. Gonzalez is rushing her to thehospital at a speed of 30.0 m/s. Witnessing the speeding car, Officer O’Malleyjumps in his police car and turns on the siren, whose frequency is 800. Hz. Ifthe officer chases after the Gonzalez’ car with a speed of 35.0 m/s, whatfrequency do the Gonzalezes hear as the officer approaches?

Answer:

12-3 Standing Waves

Waves in Strings

When a string is plucked, a wave will reflect back and forth from one end ofthe string to the other, creating nodes (points of minimum movement) andantinodes (points of maximum movement). This is called a standing wavebecause it appears to stand still.

The frequency with which a string vibrates depends upon the number ofantinodes, the wave speed, and the length of the string, as shown in thefollowing relationship.

frequency � or f �nv2L

1number of antinodes 2 1wave speed 221length 2

Waves and Sound 165

If n � 1, as shown in the diagram, the frequency is called the fundamentalfrequency. It is the lowest frequency of a vibrating string that is fixed at bothends. Multiples of the fundamental frequency are called overtones.

First overtone Second overtone Third overtone

f � � f � f � �

and so on.

Waves in Pipes

Waves in pipes that are open at both ends behave much like waves in strings.It is important to remember that antinodes always form at open ends of a pipewhile nodes form at closed ends. If a pipe is open at both ends, the possiblefrequencies are

f � (where n � 1, 2, 3 . . . for other overtones)

In a pipe that is closed at one end, the possible frequencies are

f � (where n � 1, 3, 5, 7 . . . for other overtones)

Beats

If two different frequencies sound simultaneously, the wavelengths will differ,and the crests and troughs of each wave will overlap in a way that causesvariations in loudness. There will be moments of reinforcement and momentsof cancellation as the wave patterns interact. The resulting sound is a series ofbeats, which occur when the wave sum reaches its greatest amplitude.

The beat frequency can be found by taking the absolute value of the differencebetween the two frequencies of the interacting waves.

fbeat � � f1 � f2�

nv4L

nv2L

2vL

4v2L

3v2L

vL

2v2L

166 Waves and Sound

Solved Examples

Example 6: An orchestra tunes up for the big concert by playing an A, which resoundswith a fundamental frequency of 440. Hz. Find the first and second overtonesof this note.

The first overtone is 2 times the fundamental frequency:

f2 � 2f0 so f2 � 2(440. Hz) � 880. Hz

The second overtone is 3 times the fundamental frequency:

f3 � 3f0 so f3 � 3(440. Hz) � 1320 Hz

Example 7: Zeke plucks a C on his guitar string, which vibrates with a fundamentalfrequency of 261 Hz. The wave travels down the string with a speed of 400. m/s. What is the length of the guitar string? b) Would Zeke need longeror shorter strings to play the fundamental frequency for higher notes?

a. Given: n � 1 Unknown: L � ?v � 400. m/s Original equation: f �f � 261 Hz

Solve: L � � � 0.766 m

b. If the wave speed remains the same for each string, as f gets larger, L getssmaller. Therefore, the higher the note, the shorter the string required to hearthe fundamental frequency.

Example 8: In his physics lab, Sanjiv finds that he can take a long glass tubeand fill it with water, using the air space at the top to simulate apipe closed at one end. If Sanjiv holds a tuning fork, whichvibrates with a fundamental frequency of 440 Hz, over themouth of the pipe, how long is the air column if it vibrates atthe same frequency?

Given: f � 440 Hz Unknown: L � ?v � 340.0 m/s Original equation: f �n � 1

Solve: L � � � 0.19 m11 2 1340.0 m>s 2

41440 Hz 2nv4f

nv4L

11 2 1400. m>s 221261 Hz 2

nv2f

nv2L

Waves and Sound 167

Practice Exercises

Exercise 10: Melody puts a fret on her guitar string, causing it to vibrate with afundamental frequency of 250 Hz as a wave travels through at 350 m/s. a) How long is the guitar string from the lower fixed end to the fret? b) Howfar and in which direction must the fret be moved in order to produce afundamental frequency that is twice as high (i.e., one octave higher)?

Answer: a.

Answer: b.

Exercise 11: The fundamental frequency of a bass violin string is 1045 Hz and occurs whenthe string is 0.900 m long. How far from the lower fixed end of the bass violinshould you place your fingers to allow the string to vibrate at a fundamentalfrequency 3 times as great?

Answer:

Exercise 12: Aaron blows across the opening of a partially filled 20.0-cm-high soft drinkbottle and finds that the air vibrates with a fundamental frequency of 472 Hz.How high is the liquid in the bottle?

Answer:

168 Waves and Sound

Exercise 13: A red-headed piano tuner from Chicago is tuning the Bentz’ piano when hediscovers that the G above middle C is vibrating with a higher frequency thanhis G tuning fork, which vibrates at 392.0 Hz. He plays the piano key andtuning fork at the same time and hears a beat frequency of 2.0 Hz. What is thefrequency of the G on the Bentz’ piano?

Answer:


A-1: Find the wavelength of the ultrasonic wave emitted by a bat if it has afrequency of 4.0 � 104 Hz.

A-2: Radio station KSON in San Diego broadcasts at both 1240 kHz (AM) and 97.3MHz (FM). a) Which of these signals, AM or FM, has the longer wavelength?b) How long is each?

A-3: What is the wavelength of a B note (frequency 494 Hz) played a) by a flute? b) If the flute and a sax play the same note, which of the following will bedifferent: quality, pitch, or loudness?

A-4: As an anchor is being hoisted out of the water, it hits the hull of the ship,causing the anchor to vibrate with a frequency of 150. Hz. If the speed ofsound in sea water is 1520 m/s, how many wavelengths of sound will fitbetween the boat and the ocean bottom 395 m below?

A-5: A popular pastime at sporting events is “the wave,” a phenomenon whereindividuals in the crowd stand up and sit down in sequence, causing a giantripple of people. If a continuous “wave” passes through a stadium of peoplewith a speed of 20 m/s and a frequency of 0.5 Hz, what is the distance from“crest” to “crest” (in other words, the wavelength of the wave)?

A-6: From his bedroom, Garth can hear the distant sound of a train horn as thetrain travels through the mountains on its way from Chattanooga toNashville. The horn has a frequency of 800.0 Hz as the train rolls along at20.00 m/s. What frequency does Garth hear as the train travels away?

A-7: Erin is late to physics class and is coming down the hall as the bells areringing. There are two bells in the hall, one at the far end, and one in front ofthe classroom she is approaching. Each rings with a frequency of 500.0 Hz. AsErin comes down the hallway with a speed of 1.000 m/s toward theclassroom a) what frequency does she hear for each bell? b) What beatfrequency does she hear?

Waves and Sound 169

A-8: Karen flies a motorized toy airplane with a frequency of 200. Hz in a circle ata speed of 18.0 m/s. Caroline stands nearby and hears a Doppler shift as the plane approaches and recedes from her. What are the a) highest and b) lowest frequencies Caroline hears?

A-9: Sonar detectors work by bouncing high-frequency sound waves of about 0.100 MHz off oncoming ships and detecting the frequency of the returnsignal. If a sonar detector receives a return signal of 0.101 MHz from a sub,how fast is the sub going? (Hint: Sonar travels in sea water at 1520 m/s).

A-10: A fly traveling at 3.000 m/s is pursued by a bat traveling at 6.000 m/s whoemits sound at an ultrasonic frequency of 50 000. Hz. If the fly could detectsuch a high frequency emission, what frequency would the fly hear as it isbeing pursued?

A-11: Lars is jogging beside the railroad tracks at a speed of 2.00 m/s when he hearsa train whistle behind him at a frequency of 2115 Hz. If the actual frequencyof the train whistle is 2000. Hz, how fast is the train moving?

A-12: Walter is a bass and can hit a low E that has a frequency of 82.4 Hz. Millie is asoprano and can sing as high as the third overtone of this note. What is thehighest frequency that Millie can sing?

A-13: Joyce, the church organist, is practicing on the organ and she finds that thefirst two overtones for the 370-Hz pipe are 1110 Hz and 1850 Hz. Is the organpipe closed at one end or open at both ends?

A-14: A train passes through a tunnel that is 550 m long. What is the fundamentalfrequency of vibrating air in the tunnel?

A-15: Harvey, a harpist, plucks a 0.600-m-long string on his harp.The string has a first overtone of 1046.6 Hz. How fast doesthe vibration travel through the string?

A-16: Reed arrives late to practice and finds that the orchestra hasalready tuned up and begun to play. As one oboist hits a Dwith a frequency of 293.7 Hz, Reed plays a note with afrequency of 291.2 Hz. What beat frequency is heard as thetwo instruments are playing side by side?


B-1: As a train approaches a ringing crossing gate, Stacey, a passenger on the train,hears a frequency of 440 Hz from the bell. As the train recedes, she hears afrequency of 410 Hz. How fast is the train traveling?

B-2: Richard stands on the flatbed car of a moving train playing an A on his horn.The note has a fundamental frequency of 220 Hz. Calculate whether or not thetrain could move fast enough for a stationary observer on the ground to hearthe first overtone of the horn as the train passes.

170 Waves and Sound

13 Reflection and Refraction

171

13-1 The Speed of LightAn important physical constant is the speed of light, c. In a vacuum, thisspeed is 3.00 � 108 m/s. All calculations in this book will use this value forthe speed of light unless otherwise specified in the exercise.

Light has both wave and particle properties. The exercises in this chapter dealwith the wave nature of light. For a wave of wavelength � and frequency ftraveling at the speed of light, c, c � �f. The distance that light travels in agiven amount of time can be represented by the equation �d � c�t.

Note that these two equations are both special cases of the more generalequations, v � �f and �d � v�t.

Solved Examples

Example 1: How long does it take for light from the sun to reach Earth if the sun is 1.50 � 1011 m away?

Given: �d � 1.50 � 1011 m Unknown: �t � ?c � 3.00 � 108 m/s Original equation: �d � c�t

Solve: �t � � � 500. s

This is a little more than 8 min.

Example 2: Microwave ovens emit waves of about 2450 MHz. What is the wavelength ofthis light?

Solution: The term MHz stands for Megahertz or 106 Hz. Therefore, themicrowaves have a frequency of 2450 � 106 Hz.

Given: c � 3.00 � 108 m/s Unknown: � � ?f � 2450 � 106 Hz Original equation: c � �f

Solve: � � � � 0.122 m3.00 � 108 m>s2450 � 106 Hz

cf

1.50 � 1011 m3.00 � 108 m>s

¢dc

Practice Exercises

Exercise 1: When you look at a distant star or planet, you are looking back in time. Howfar back in time are you looking when you observe Pluto through the telescope from a distance of 5.91 � 1012 m?

Answer:

Exercise 2: If a person could travel at the speed of light, it would still take 4.3 years toreach the nearest star, Proxima Centauri. How far away, in meters, is ProximaCentauri? (Ignore any relativistic effects.)

Answer:

Exercise 3: When you go out in the sun, it is the ultraviolet light that gives you your tan.The pigment in your skin called melanin is activated by the enzyme tyrosinase,which has been stimulated by ultraviolet light. What is the wavelength of thislight if it has a frequency of 7.89 � 1014 Hz?

Answer:

Exercise 4: IRAS, the Infrared Astronomy Satellite launched by NASA in 1983, had adetector that was supercooled to enable it to measure infrared or heatradiation from different regions of space. What is the frequency of infraredlight that has a wavelength of 1.00 � 10�6 m?

Answer:


13-2 ReflectionVocabulary Reflection: The bouncing of light.

The angle a beam of light makes when it strikes a surface is described with respect to the normal, animaginary line drawn perpendicular to the surface.When light shines onto a mirror, the angle at whichthe light enters the mirror (angle of incidence) isexactly equal to the angle at which the light leavesthe mirror (angle of reflection). This is called thelaw of reflection and is easily observed in a plane(flat) mirror.

Due to the curvature of a spherical mirror, lightreflected from its surface behaves somewhatdifferently than it does when reflected from aplane mirror. There are two types of sphericalmirrors, converging (or concave) and diverging(or convex).

Converging Diverging

The following terminology is used when describing how light is reflectedfrom converging and diverging mirrors.

Vocabulary Object distance: The distance from the mirror to the object. This value isalways a positive number.

Vocabulary Image distance: The distance from the mirror to the image. An image can bereal (inverted and able to be projected on a screen), or virtual (right-side-upand not able to be projected on a screen).

Vocabulary Focal point: The point where parallel rays meet (or appear to meet) afterreflecting from a mirror. The distance from this focal point to the mirror iscalled the focal length. The focal length of a converging mirror always has apositive value while the focal length of a diverging mirror always has anegative value.

Vocabulary Mirror Equation: � �

� �

Note: Many situations involving mirrors can also be solved using raydiagrams.

1di

1do

1f

1image distance

1object distance

1focal length

Reflection and Refraction 173

Converging (Concave) Mirror

If an object is located more than one focal length from a converging mirror asshown in Figure A, the image it forms is real, inverted, and in front of themirror. You can actually project this image onto a piece of paper. Both do anddi have positive values.

If the object is at the focal point as in figure B, no image is formed because thereflected rays are parallel.

If an object is located less than one focal length from a converging mirror as infigure C, the image it forms is virtual, upright, enlarged, and behind themirror. In other words, you must look into the mirror to see the image. Here,do has a positive value and di has a negative value.

Figure A Figure B Figure C

Diverging (Convex) Mirror

The image formed by a diverging mirror isalways virtual, upright, smaller, and behindthe mirror. The image can be seen only bylooking into the mirror. Here do has apositive value while di has a negativevalue.

Solved Examples

Example 3: Sitting in her parlor one night, Gerty sees the reflection of her cat, Whiskers,in the living room window. If the image of Whiskers makes an angle of 40°with the normal, at what angle does Gerty see him reflected?

Solution: Because the angle of incidence equals the angle of reflection, Gertymust see her cat reflected at an angle of 40°.

Example 4: Wendy the witch is polishing her crystal ball.It is so shiny that she can see her reflection whenshe gazes into the ball from a distance of 15 cm. a) What is the focal length of Wendy’s crystal ballif she can see her reflection 4.0 cm behind thesurface? b) Is this image real or virtual?


a. Given: do � 15 cm Unknown: f � ?di � �4.0 cm Original equation: � �

Solve: � � � �

Getting a common denominator of 60 cm gives � � �

To find f, take the reciprocal of this sum. f � � �5.5 cm

The minus sign before the answer means that this is the focal length of adiverging mirror.

b. The image seen behind a curved surface is always a virtual image.

Example 5: With his face 6.0 cm from his empty water bowl, Spot sees his reflection 12 cmbehind the bowl and jumps back. a) What is the focal length of the bowl?b) What was surprising about Spot’s reflection that may have caused him tojump?

a. Given: do � 6.0 cm Unknown: f � ?di � �12 cm Original equation: � �

Solve: � � � �

Getting a common denominator of 12 cm gives � � �

f � 12 cm

The positive answer means that the bowl was acting as a converging mirror.

b. The surprising thing Spot noticed about his reflection was that it appearedlarger than life!

Practice Exercises

Exercise 5: Manish is in a house of mirrors with one of his friends when he comes to twomirrors situated at an angle of 90°. Manish stands so that light shining on hisface is incident on one mirror at an angle of 50°, as shown. At what angle willthis light reflect from the second mirror?

Answer:

112 cm

112 cm

212 cm

1f

1�12 cm

16.0 cm

1di

1do

1f

1di

1do

1f

�60 cm11

�1160 cm

1560 cm

460 cm

1f

1�4.0 cm

115 cm

1di

1do

1f

1di

1do

1f


Exercise 6: A popular lawn ornament in the 1960s was a colored reflecting sphere that satin the yard as a decoration. a) If a bird is 10.0 cm from a blue reflecting sphereand sees its image reflected 5.0 cm behind the sphere, what is the focal lengthof the spherical reflector? b) Would the bird’s image appear larger or smallerthan the bird itself?

Answer: a.

Answer: b.

Exercise 7: Polly applies her mascara while looking in a concave mirror whose focallength is 18 cm. She looks into it from a distance of 12 cm. a) How far isPolly’s image from the mirror? b) Does it matter whether or not Polly’s face iscloser or farther than one focal length? Explain.

Answer: a.

Answer: b.

Exercise 8: A friend is wearing a pair of mirrored sunglasses whose convex surface has afocal length of 20.0 cm. If your face is 40.0 cm from the sunglasses, how farbehind the sunglasses is your image?

Answer:


13-3 RefractionVocabulary Refraction: The change in direction of light due

to a change in speed as it passes from onemedium to another.

The path of light is described with respect to thenormal. If light is slowed down as it enters anew medium, it bends toward the normal. If itspeeds up, it bends away from the normal.

The amount of bending is represented with theletter n, which stands for the index of refraction. The index of refraction for aparticular medium is a ratio of the speed of light in a vacuum to the speed oflight in the medium.

index of refraction � or n �

Because light travels fastest in a vacuum, the index of refraction for any othermedium is always greater than 1. Although the index of refraction for air is1.0003, in this chapter the value will be written simply as 1.00.

The angle to which light will bend upon passing from one medium to anotherdepends upon the index of refraction of each of the two media, n1 and n2, andthe light’s angle of incidence.

n1 sin �1 � n2 sin �2

The symbols �1 and �2 stand for the angle of incidence and the angle ofrefraction, respectively.

A special case of this equation is used when light travels from a more-densemedium to a less-dense medium and the refracted ray makes an angle of 90.0°with the normal as it skims along the boundary of the two media. When thishappens, the incident angle �1 is called the critical angle, �c.

n1 sin �c � n2 sin 90.0°

If the incident angle is any bigger than the critical angle, there is no refraction.Instead, all the light is reflected back inside the object. This is called totalinternal reflection.

cv

speed of light in a vacuum

speed of light in another medium


Solved Examples

Example 6: Hickory, a watchmaker, is interested in an old timepiece that’s been broughtin for a cleaning. If light travels at 1.90 � 108 m/s in the crystal, what is thecrystal’s index of refraction?

Given: c � 3.00 � 108 m/s Unknown: n � ?v � 1.90 � 108 m Original equation: n �

Solve: n � � � 1.58

Remember, the index of refraction has no units. It is just a ratio of the speed oflight in two different media.

Example 7: While fishing out on the lake one summerafternoon, Amy spots a large trout just belowthe surface of the water at an angle of 60.0° tothe vertical, and she tries to scoop it out of thewater with her net. a) Draw the fish whereAmy sees it. b) At what angle should Amy aimfor the fish? (nwater � 1.33).

Solution: a. The fish will appear to be straightahead according to Amy. However, becauselight travels slower in water than in air, the fishis closer to Amy than she thinks.

b. Given: n1 � 1.33 (water) Unknown: �2 � ?n2 � 1.00 (air) Original equation: n1 sin �1 � n2 sin �2�2 � 60.0°

Solve: sin �1 � � � 0.651 �1 � sin�1 0.651 � 40.6°

Example 8: Binoculars contain prisms inside that reflect light entering at an angle largerthan the critical angle. If the index of refraction of a glass prism is 1.58, whatis the critical angle for light entering the prism?

Given: n1 � 1.58 (glass) Unknown: �c � ?n2 � 1.00 (air) Original equation: n1 sin �c � n2 sin 90.0°

Solve: sin �c � � � 0.633 �c � sin�1 0.633 � 39.3°11.00 2 sin 90.0°

1.58n2 sin u2

n1

11.00 2 sin 60.0°1.33

n2 sin u2n1

3.00 � 108 m>s1.90 � 108 m>s

cv

cv


Practice Exercises

Exercise 9: Alison sees a coin at the bottom of her swimming pool at an angle of 40.0° tothe normal and she dives in to retrieve it. However, Alison doesn’t like toopen her eyes in the water so she must rely on her initial observation of thecoin made in the air. At what angle does the light from the coin travel as itmoves toward the surface? (nwater � 1.33)

Answer:

Exercise 10: Here’s an interesting trick to try. Place a penny in the bottom of a cup andstand so that the penny is just out of sight, as shown. Then pour water intothe cup. Without moving, you will suddenly see the penny magically appear.If you look into the cup at an angle of 70.0° to the normal, at what angle to thenormal must the penny be located in order for it to just appear in the bottomof the cup when the cup is filled with water? (nwater � 1.33)

Answer:

Exercise 11: Rohit makes his girlfriend a romantic candlelight dinner and tops it off with adessert of gelatin filled with blueberries. If a blueberry that appears at anangle of 44.0° to the normal in air is really located at 30.0° to the normal in thegelatin, what is the index of refraction of the gelatin?

Answer:


Exercise 12: A jeweler must decide whether the stone in Mrs. Smigelski’s ring is a realdiamond or a less-precious zircon. He measures the critical angle of the gemand finds that it is 31.3°. Is the stone really a diamond or just a goodimitation? (ndiamond � 2.41, nzircon � 1.92)

Answer:


A-1: Radio waves travel at the speed of light. How long would it take the Russiansto send a message to a spacecraft orbiting Mars at a distance of 7.8 � 1010 mfrom Earth?

A-2: At the doctor’s office, an X-ray of your hand is taken with electromagneticradiation of frequency 3.00 � 1017 Hz. What is the wavelength of this radiation?

A-3: In order to see your back teeth more easily,your dentist uses a small mirrored instrumentthat can be easily manipulated in your mouth.If the dentist places this mirror directly undera real molar, and tilts it 20°, at what angle tothe normal will the dentist need to look intothe mirror in order to see the tooth?

A-4: While decorating his Christmas tree, Vinnie discovers that he can see hisreflection in a Christmas tree ball. a) If Vinnie looks into the ornament from adistance of 20.0 cm and focuses on his reflection 4.0 cm behind the ball, whatis the focal length of the Christmas ball? b) Is Vinnie’s image upright orinverted? c) Is his image larger or smaller?

A-5: Some rear-view mirrors on cars and trucks are curved to allow for a widerfield of view. a) Would these mirrors be converging or diverging? b) Whymight this be a little dangerous for a driver unaccustomed to this type ofmirror? c) If the mirror has a focal length of 20.0 cm and the truck driver looksin the mirror from a distance of 30.0 cm, where does he see his image?

A-6: Wes stands in his hotel room in Cancun and admires his tan in a mirror thatallows him to look “larger than life.” a) What type of mirror is Wes using? b) Where should Wes stand in relation to the focal point of the mirror in orderto appear enlarged? c) If the mirror has a focal length of 75.0 cm, and Wesstands 50.0 cm from the mirror’s surface, how far behind the mirror is hisimage? d) Where does he see his image if he stands 200. cm from the mirror?


A-7: An automobile headlight is made by placing a filament at the focal point of aconcave mirrored surface. a) If the focal length of the mirrored surface is 5.0 cm, calculate the image distance. b) Why is this the desired image distancefor automobile headlights?

A-8: A blue glow from a bug light strikes the Bradford’s swimming pool at anangle of 35.0°. At what angle is the light refracted into the pool?(nwater � 1.33)

A-9: The index of refraction of ethyl alcohol is 1.36, while the index of refraction ofwater is 1.33. a) Does light travel faster in alcohol or in water? b) What is thespeed of light in each?

A-10: Heather is snorkeling in Oahu’s Hanuma Baywhen she looks up through the water at a palmtree on the shore. a) If the index of refraction ofwater is 1.33 and Heather sees the palm tree atan angle of 45°, at what angle is the palm treereally located with respect to the normal?

A-11: Spenser, a cat, enjoys watching the familygoldfish from the top of the fish tank. If thegoldfish, swimming in water, appears to be atan angle of 28.0° as seen by Spenser, at what true angle is the goldfish fromthe normal? (nwater � 1.33)

A-12: Evan has taken Eva out to dinner to propose marriage and he has hidden theengagement ring in her drink as a surprise. When Eva has finished her drink,she spots the ring beneath an ice cube. If Eva looks down into the glass at anangle of 61.0° but the ice cube refracts the ring at an angle of 42.0°, what is theindex of refraction of ice?

A-13: In her bedroom, Mia has a fiber optic light that glows as hundreds of fiberoptic cables are lit from below. a) If each fiber optic cable has an index ofrefraction of 1.48, at what critical angle must light enter the cable in order fortotal internal reflection to occur? b) Explain why total internal reflection isimportant to a fiber optic lamp.


B-1: Marian admires a new dress in adepartment store dressing room mirror.If Marian stands as shown, making anangle of 70° with the center mirror, atwhat angle will the light be reflectedfrom the mirror on the right?


B-2: Your friend is stranded 10.0 m high in a tall treewith a hungry tiger beneath, while you lie on thebeach a distance away. He has only a mirror, whichhe uses to signal you by holding it perpendicular tothe horizon as shown. If the sun hits the mirror at a30.0° angle to the normal and reflects back in youreye, how far away are you from the tree?

B-3: As you are walking toward a swimming pool on a hot summer day, yousuddenly notice a glare of sunlight off the water’s surface that is so bright itmakes you close your eyes. If the angle of incidence of the incoming sunlightis 70.0° and you stand 1.80 m tall, how far (horizontally) are you standingfrom the point where the incident ray hits the water?

B-4: The deepest section of ocean in the world is the Marianas Trench, located inthe Pacific Ocean. Here, the ocean floor is as low as 10 918 m below thesurface. If the index of refraction of water is 1.33, how long would it take alaser beam to reach the bottom of the trench?


14 Lenses, Diffraction,and Interference

183

14-1 Lenses, Telescopes, and Magnifying GlassesWhen light shines through a lens, it is refracted or bent due to the shape andmaterial of the lens. Parallel rays of light passed through some lenses willeventually converge at the focal point. The terminology used for lenses ismuch the same as that used for mirrors in Chapter 13.

Vocabulary Object distance: The distance from the center of the lens to the object.

Vocabulary Image distance: The distance from the center of the lens to the image. Animage can be real (able to be projected on a screen), or virtual (not able to beprojected on a screen).

Vocabulary Focal point: The point where parallel rays meet (or appear to meet) afterpassing through a lens. The distance from this focal point to the center of thelens is called the focal length.

Thin Lens Equation: � �

or � �

NOTE: Many situations involving lenses can also be solved using raydiagrams.

The Converging (Positive) Lens

The focal length of a converging lens isalways a positive number.

If an object is located outside the focalpoint of a converging lens, the image itforms is real, inverted, and on theopposite side of the lens. Both do and diare positive numbers.

1di

1do

1f

1image distance

1object distance

1focal length

If an object is located inside the focalpoint of a converging lens, the image itforms is virtual, upright, enlarged, andon the same side as the object. In thisinstance, do is positive and di is negative.

If the object is at the focal point, the raysdo not converge and therefore no imageis formed.

The Diverging (Negative) Lens

The focal length of a diverging lens isalways a negative number.

The image formed by a diverging lens isalways virtual, upright, smaller, and onthe same side of the lens as the object. Inthis instance, do is positive and di isnegative.

If an object appears taller when seen through a lens, the object is magnified.The linear magnification of an object can be found by comparing the imagedistance to the object distance, or by comparing the image height, hi, to theobject height, ho.

linear magnification � �

or m � �

Note that a negative magnification implies a virtual image.

Linear magnification has no units. It is simply a ratio of image to objectdistance or a ratio of image to object height.

The Refracting Telescope

A refracting telescope is a device that uses one lens to produce a real image,and a second lens to produce the virtual image that is seen by your eye. Theamount of linear magnification you see when you look at an object through atelescope depends upon the focal length of each of the lenses. The lens thatpoints toward the object is the objective lens and the lens you look through isthe eyepiece. The focal lengths of each of these lenses are labeled fo and fe,respectively.

hi

ho

di

do

image height

object height

image distance

object distance

184 Lenses, Diffraction, and Interference

linear magnification � or m �

The Magnifying Glass

When using a magnifying glass, the amount of angular magnification of anobject depends upon how close you hold the magnifying glass to the object. Italso depends upon the near point of your own eye, which is the closest pointat which an unaided eye can focus on an object. A person’s near pointincreases with age and the eyes lose some of their adaptable, elasticproperties. However, for the ease of calculations, assume the near point of theeye is 25 cm unless otherwise noted.

angular magnification � or M �

Solved Examples

Example 1: Mukluk, an Inuit, makes a converging lens out of ice that will enable him toconcentrate light from the sun to start a fire. When he holds the ice lens 1.00 m from a snow-covered wall, an image of his 5.00-m-distant igloo isprojected onto the snow. a) What is the focal length of the ice lens? b) Draw a ray diagram of the situation.

a. Given: do � 5.00 m Unknown: f � ?di � 1.00 m Original equation: � �

Solve: � � � � � 1.20 m�1

Taking the reciprocal gives f � � 0.833 m

The focal length of 0.833 m is close to the image distance of 1.00 m.

b.

Example 2: A diverging lens is placed 5.0 cm in front of a laser beam to spread the lightfor the production of a hologram. a) What is the focal length of the lens if thebeam of laser light seems to come from a point 2.0 cm behind the lens? b) Draw a ray diagram of the situation.

a. Given: do � 5.0 cm Unknown: f � ?di � �2.0 cm Original equation: � �

1di

1do

1f

11.20 m�1

11.00 m

15.00 m

1di

1do

1f

1di

1do

1f

near point

fnear point

focal length

fofe

focal length of objective lens

focal length of eyepiece

Lenses, Diffraction, and Interference 185

Solve: � � � � � � � � cm�1

f � � cm � �3.3 cm

b.

Example 3: Irwin, a coin collector, is looking at a rare coin held behind a magnifying glasswhose focal length is 5.0 cm. a) If the eyes’ near point is 25 cm, what is theangular magnification? b) If the coin is 2.0 cm in diameter, how large will itsdiameter appear to be when it is held in this position under the magnifyingglass?

a. Given: near point � 25 cm Unknown: M � ?f � 5.0 cm Original equation: M �

Solve: M � � � 5.0 The coin is magnified 5.0 times.

b. Given: m � 5.0 Unknown: hi � ?ho � 2.0 cm Original equation: m �

Solve: hi � mho � (5.0)(2.0 cm) � 10. cm

Example 4: The ship Speedwell brought many early settlers to this country in the 1600s.Oceanus sits high above the ship’s deck in the crow’s nest watching through atelescope for the first sign of land. How much does the telescope magnify ifthe eyepiece has a 2.0-cm focal length and the objective lens has a 80.-cm focallength?

Given: fo � 80. cm Unknown: m � ?fe � 2.0 cm Original equation: m �

Solve: m � � � 40. The telescope magnifies 40. times.

Practice Exercises

Exercise 1: Harold and Roland find a discarded plastic lens lying on the beach. The boysdiscuss what they learned in physics last semester and argue whether the lensis a converging or a diverging one. When they look through the lens, theynotice that objects are inverted. a) If an object sitting 25.0 cm in front of thelens forms an image 20.0 cm behind the lens, what is the focal length of thelens? b) Is it a converging or a diverging lens?

80. cm2.0 cm

fofe

fofe

hi

ho

25 cm5.0 cm

near point

f

near point

f

10.3

310.

510.0 cm

210. cm

1�2.0 cm

15.0 cm

1di

1do

1f


Answer: a.

Answer: b.

Exercise 2: Sadie looks at her friend’s face through a diverging lens. a) Is the image realor virtual? b) If her friend’s face is 50.0 cm from the lens that forms an imageat a distance of 20.0 cm, what is the focal length of the lens? c) Draw a raydiagram of the situation.

Answer: a.

Answer: b.

Exercise 3: Giorgio is clicking shots of the fashion model Nadine as she walks towardhim across the studio. Giorgio’s camera contains a lens with a focal length of0.0500 m. a) How far back must the film be located when Nadine is 3.00 mfrom the camera? b) Should the lens be moved in or out as Nadine approachescloser to the photographer? c) Draw a ray diagram of the situation withNadine at 3.00 m and 1.00 m from the camera.

Answer: a.

Answer: b.

Exercise 4: Dr. Wasserman is showing slides to his biology class. a) If the slides arepositioned 15.5 cm from the projector lens that has a focal length of 15.0 cm,where should the screen be placed to produce the clearest image of the slide?b) Draw a ray diagram of the situation.

Answer: a.


Exercise 5: Marlin is out on a safari. Looking through his telescope, he spots a giraffe inthe distance. The telescope has an objective lens of 40-cm focal length and aneyepiece of 2-cm focal length. a) What is the magnification of the giraffe?b) How large is the image formed by the telescope if the giraffe appears to be1.5 cm high to the naked eye?

Answer: a.

Answer: b.

Exercise 6: Emilio, an entomologist, studies a millepede that he holds behind amagnifying glass whose focal length is 2.00 cm. a) Assuming Emilio’s nearpoint is 25.0 cm, what is the angular magnification? b) Does Emilio have tobring the magnifying glass closer to, or farther from, the millipede in order tomake it appear larger?

Answer: a.

Answer: b.

Exercise 7: Mr. Crabtree, a jeweler, looks through his jeweler’s loupe (a small magnifyingglass attached to his glasses) in order to read the engraving on a pewter bowl.The loupe has a focal length of 3 cm. If Mr. Crabtree’s near point is 24 cm,what is the angular magnification of the engraving?

Answer:


14-2 EyeglassesWhen the eye is unable to focus incoming light directly on the retina (a layerof tissue in the back of the eye that is sensitive to light), eyeglasses or contactlenses are usually prescribed.

If the lens, or cornea, is curved so that light would focus behind the retina, theresult is a condition called farsightedness, where only objects at a distancecan be seen clearly. To correct this problem, glasses for a farsighted personhave lenses that are thicker in the middle and thinner near the edges(converging lenses).

If the lens, or cornea, is curved so that light would focus in front of the retina,the result is a condition called nearsightedness, where only objects close upcan be seen clearly. To correct this problem, glasses for a nearsighted personhave lenses that are thinner in the middle and thicker near the edges(diverging lenses).

The power of a pair of prescription glasses is the reciprocal of the focal length,if the focal length is measured in meters.

Power � or P �

The SI unit for the power of eyeglasses is the diopter, which equals thereciprocal of a meter (m�1).

For all the following exercises, assume that the preferred far point of the eye isinfinity, , and the preferred near point is 25 cm. To find the power of thelenses in a pair of glasses, take the difference between the reciprocal of howfar the eye can see without glasses and how far it can see with glasses.

power � � �

If you wear glasses or contact lenses, ask your doctor about the power of yourprescription. You may find that it can be different for each eye!

1do1no glasses2

1do1glasses2

1fglasses

1f

1focal length


Solved Examples

Example 5: Craig is nearsighted, so he must wear glasses to see objects that are far away.If his glasses have a focal length of 0.5 m, what is their power in diopters?

Solution: The focal length must be written as a negative number because anearsighted person will always wear glasses with diverging lenses. Adiverging lens has a negative focal length.

Given: fglasses � �0.5 m Unknown: P � ?Original equation: P �

Solve: P � � � �2 diopters

Example 6: In the previous exercise, if Craig can see to infinity with his glasses on, whatis the maximum distance he can see clearly with the glasses off?

Given: fglasses � �0.5 m Unknown: do(no glasses) � ?do(glasses) � Original equation:

� �

Solve: � � � � � 0 � (�2) � 2 diopters

do(no glasses) � � 0.5 m

The farthest Craig can see clearly without glasses is 0.5 m.

Example 7: Dorcas must hold the phone book 0.5 m from her eyes in order to find the eyedoctor’s phone number. a) If Dorcas would like to read the phone book at amore comfortable distance of 0.25 m, what power glasses does she need? b) What type of lenses would these glasses contain?

a. Given: do(no glasses) � 0.5 m Unknown: P � ?do(glasses) � 0.25 m Original equation:

� �

Solve: � � � � � 4 � 2 � 2 diopters

b. Because the power of the glasses in this example is a positive number, thelenses must be converging lenses. This is supported by the fact thatfarsightedness must be corrected with converging lenses.

10.5 m

10.25 m

1do1no glasses2

1do1glasses2

1fglasses

1do1no glasses2

1do1glasses2

1fglasses

12 diopters

1�0.5 m

1q

1fglasses

1do1glasses2

1do1no glasses2

1do1no glasses2

1do1glasses2

1fglasses

1�0.5 m

1f

1f


Practice Exercises

Exercise 8: Beth is farsighted, so she must wear glasses to see objects close by. If herglasses have a focal length of 0.30 m, what is their power in diopters?

Answer:

Exercise 9: Herman is able to read the newspaper at a distance of 0.75 m, but no closer. a) Is he farsighted or nearsighted? b) What power lens should he use to allowhim to read the paper at 0.25 m? c) What type of lens does he need?

Answer: a.

Answer: b.

Answer: c.

Exercise 10: At the beach, Maria can see Sandy, a surfer, clearly only when he is standingcloser than 2.0 m. a) What power prescription sunglasses would Maria needin order to see Sandy when he is out on the ocean riding a wave? b) Whattype of lenses will her glasses contain?

Answer: a.

Answer: b.


Exercise 11: Matt is driving his “18-wheeler” while wearing his new pair of glasses whosefocal length is �0.40 m. If the glasses allow Matt to see clearly at an infinitedistance for normal driving, how far could Matt see clearly before he boughtthe glasses?

Answer:

Exercise 12: Moshe has gone to Bermuda for spring vacation and when he is on the beachrealizes that he has picked up his father’s pair of prescription sunglasses bymistake. The glasses have a power of � 3.0 diopters. a) What type of eyeproblem does Moshe’s father have, and how do you know? b) What is theclosest that Moshe’s father can see clearly without his glasses? c) Will theseglasses produce an image in front of, or behind, the image formed by Moshe’snormal eye?

Answer: a.

Answer: b.

Answer: c.

14-3 Diffraction and InterferenceVocabulary Diffraction: The spreading of a wave as it passes around an obstacle or

through an opening.

Vocabulary Interference: When two waves overlap to produce one new wave.

In 1801, Thomas Young attempted to provethat light was a wave by showing that it hasthe ability to diffract and interfere. Youngpassed white light through two closely-spacedslits and noticed that the light spread out as itpassed through the openings (diffracted), andoverlapped on a screen a few meters away(interfered), to produce alternating bands oflight and dark.


Whether light is passed through two slits or through the multiple, closely-spaced slits of a diffraction grating, the grating equation can be written as

wavelength � or � �

This equation is a good approximation when the angular separation betweenthe bright bands is very small. When used with a diffraction grating, however,it could produce an answer with as much as 10% error. Nevertheless, tosimplify calculations and avoid the use of trigonometry, the equation will beused in this form in all exercises.

The common unit for the wavelength of light is the nanometer (nm), whichequals 10�9 m.

Solved Examples

Example 8: Miss McGillivray loses the specifications for her diffraction grating and mustrecalibrate it in order to determine the grating spacing. She shines a redhelium-neon laser, whose wavelength is 633 nm, through the grating. Twobright spots that are each 1.40 m from the central maximum fall on the wall4.00 m away. What is the space between the grooves on the diffraction grating?

Solution: First, convert nm to m. 633 nm � 6.33 � 10�7 m

Given: � � 6.33 � 10�7 m Unknown: d � ?L � 4.00 m Original equation: � �x � 1.40 m

Solve: d � � � 1.81 � 10�6 m

Example 9: In the previous exercise, Miss McGillivray uses her newly calibrated gratingto determine the wavelength of a green helium-neon laser. Keeping the laserat the same distance from the wall as before, the distance from the centralmaximum to the first bright fringe is 1.20 m. What is the wavelength of thegreen HeNe laser?

Given: d � 1.81 � 10�6 m Unknown: � � ?L � 4.00 m Original equation: � �x � 1.20 m

Solve: � � � � 5.43 � 10�7 m � 543 nm11.81 � 10�6 m 2 11.20 m 2

4.00 mdxL

dxL

16.33 � 10�7 m 2 14.00 m 21.40 m

lLx

dxL

dxL

1slit separation 2 1space between bright bands 21distance from slits to screen 2


Practice Exercises

Exercise 13: Judy and Earl are sitting under the boardwalk one warm summer eveningwhile the light of a low-pressure sodium vapor lamp whose wavelength is 589 nm passes through two small cracks in a board, producing fringes of light0.0020 m apart on the ground. a) If the boardwalk is 3.0 m above the sand,what is the distance between the two cracks in the board? b) If the distancebetween the cracks were smaller, would the fringes of light on the ground becloser together or farther apart?

Answer: a.

Answer: b.

Exercise 14: Two large speakers broadcast the sound of a band tuning up before anoutdoor concert. While the band plays an A whose wavelength is 0.773 m,Brenda walks to the refreshment stand along a line parallel to the speakers. Ifthe speakers are separated by 12.0 m and Brenda is 24.0 m away, how farmust she walk between the “loudspots”?

Answer:

Exercise 15: In an attempt to test the particle nature of matter, Claus Jönsson performed anexperiment in 1961 that was very similar to Young’s Double Slit experimentfor light done in 1801. Jönsson sent a beam of electrons through two slitsseparated by 2.00 � 10�6 m onto a fluorescent screen 0.200 m away. Due to their high speed, the electrons behaved like waves with a wavelength of2.40 � 10�11 m. How far apart were the bright lines formed on the screen?

Answer:



A-1: A photocopy machine is set to reduce the size of printed material by 50%.When the print is regular size, both the image and object distance are 16.0 cm.If the lens is then moved 24.0 cm from the object, how large is the new imagedistance?

A-2: The average normal human eye forms an image on the retinaat a distance of about 0.0240 m from the lens, as shown. Howmuch must the focal length of the lens change in order toaccommodate an object moved from 10.0 m to 0.250 m? (Thischange in focal length is accomplished by small muscles in theeye called cilliary muscles. These muscles actually stretch andrelax the lens.)

A-3: Lisa is posing for her senior class picture and sits 2.00 m from the camera lenswhose focal length is 17.0 cm. The camera lens is positioned 21.0 cm in frontof the film. Will the photographer obtain a clear image of Lisa? If not, by howmuch must the camera lens be moved in our out?

A-4: Cindy is lying on the beach focusing her camera on a friend standing 5.00 maway. Her camera has a focal length of 5.00 cm. a) Where must Cindy positionthe camera lens relative to the film for the sharpest focus? b) What type oflens must her camera have, and why?

A-5: Sherlock Holmes discovers some telltale hairs at the scene of a crime. Heviews the hairs with his magnifying glass from a distance of 6.0 cm. If thehairs are magnified 4.0 times, how far is the magnified image from the lens?

A-6: Jacob attaches a solar filter to his telescope and projects an image of the sunthrough the objective lens that has a focal length of 2.00 m. Jacob can’t decidewhether to use a 40.0-mm eyepiece or a 16.0-mm eyepiece to study the solarfeatures. a) What amount of magnification will each eyepiece provide? b) Someone may look through a telescope and ask, “What is the magnificationof this instrument?” Why is it impossible to give one standard answer to thequestion? c) If the sun appears to be 1.00 cm across to the naked eye, howlarge will it appear when viewed with the 16.0-mm eyepiece?

A-7: To the naked eye, Jupiter appears to be about 0.10 cm in diameter. In atelescope whose objective lens has a focal length of 2.0 m, Jupiter appears tobe 1.2 cm in diameter. What is the focal length of the eyepiece used to producethis image?

A-8: Ms. Chang is standing by the slide projector in the back of the room when sherealizes that the screen is in the wrong location to get a clear image. a) If theprojector has a lens with a focal length of 20.0 cm, and the slides sit 20.6 cmbehind the lens, in which direction should one of the students move thescreen that sits 7.00 m from the lens? b) How far away should the screen befrom the projector lens?


A-9: Beverly wears bifocals. She can read close up when she looks through thebottom portion and can read far away when she looks through the topportion. a) The top of her glasses has a focal length of �0.25 m. What is thepower, in diopters, of this part of the glasses? b) The bottom portion has apower of 3.5 diopters. What is the focal length of this part of the glasses?

A-10: In exercise A-9, if Beverly can see to infinity with her glasses on, a) what is themaximum distance she can see clearly with the glasses off? b) If Beverly cansee an object at 25 cm with her glasses on, what is the minimum distance shecan see clearly with the glasses off?

A-11: Rachel brings a note home from school. The note advises her mother that“Rachel is having a difficult time reading the words on the board and canonly see the words if she is sitting closer than 2.0 m.” If Rachel wants to beable to read the words from 3.0 m away, what power glasses does she need?

A-12: Joon puts on a pair of diffraction grating glasses that he bought in a noveltyshop and looks at a mercury vapor street lamp that is 5.00 m away. He sees ayellow spectral line 1.16 m on either side of the light source. If the diffractiongrating glasses have a slit separation of 2.49 � 10�6 m, what is thewavelength of the light Joon is observing?

A-13: Radio station WLLH has two transmitters that sit atop nearby hillsidesbroadcasting a wave that is 214 m long. As Kiesha drives down the interstateparallel to the two transmitters at a distance of 1000. m, she hears an increasein signal from the station every 30.0 m. How far apart are the twotransmitters?


B-1: The Hale telescope at the Yerkes Observatory in Wisconsin has an objectivelens with a focal length of 19 m. (For an object at infinity, the image distanceequals the focal length.) If the telescope is used to observe Saturn that is1275 � 109 m from Earth, what will be the apparent diameter of the rings iftheir actual diameter is 27 � 107 m?

B-2: Dr. Kirwan is preparing a slide show that he will present to the executiveboard at tonight’s committee meeting. He places a 3.50-cm slide behind a lensof 20.0 cm focal length in the slide projector. a) How far from the lens shouldthe slide be placed in order to shine on a screen 6.00 m away? b) How widemust the screen be to accommodate the projected image?

B-3: Madeline is working for the Eye-Spy Detective Agency and her assignment isto secretly photograph the pages of a journal. Madeline’s tiny camera has thefilm located 2.10 cm behind the lens, and she must fill the entire piece of 1.00-cm film with the picture of the 25.0-cm-tall document. How close mustMadeline be to the journal pages to get a clear image on the film?


15 Electrostatics

197

15-1 Electrostatic ForceVocabulary Electrostatics: The study of electric charges, forces, and fields.

The symbol for electric charge is the letter “q” and the SI unit for charge is thecoulomb (C). The coulumb is a very large unit.

1 C � 6.25 � 1018 electrons or1 electron has a charge of 1.60 � 10�19 C.

Electrons surrounding the nucleus of an atom carry a negative charge.Protons, found inside the nucleus of the atom, carry a positive charge of1.60 � 10�19 C, while neutrons (which also reside in the nucleus) are neutral.It is important to remember that only electrons are free to move in asubstance. Protons and neutrons usually do not move.

When two objects with like charges, positive or negative, are brought neareach other, they experience a repulsive force. When objects with oppositecharges, one negative and one positive, are brought side by side, theyexperience an attractive force. These forces can be described with Coulomb’s law.

Vocabulary Coulomb’s Law: Two charged objects attract each other with a force that isproportional to the charge on the objects and inversely proportional to thesquare of the distance between them.

F �

This equation looks very similar to Newton’s law of universal gravitation. Asbefore, the sign � means “proportional to.” To make an equation out of thisproportionality, insert a quantity called the electrostatic constant, k.

k � 9.0 � 109 N � m2/C2

The magnitude of Coulomb’s law can now be written as an equation.

electrostatic force � or F �kq1q2

d2(electrostatic constant)(charge 1)(charge 2)

(distance)2

q1q2

d2

Like all other forces, the electrostatic force between two charged objects ismeasured in newtons.

Solved Examples

Example 1: Anthea rubs two latex balloons against her hair, causing the balloons tobecome charged negatively with 2.0 � 10�6 C. She holds them a distance of0.70 m apart. a) What is the electric force between the two balloons? b) Is itone of attraction or repulsion?

Solution: It is not necessary to carry the sign of the charge throughout theentire exercise. However, when determining the direction of your finalanswer, it is important to remember the charge on each object.

Given: q1 � 2.0 � 10�6 C Unknown: F � ?q2 � 2.0 � 10�6 C Original equation: F �d � 0.70 mk � 9.0 � 109 N � m2/C2

Solve: F � � � 0.073 N

b) Because both balloons are negatively charged, they will repel each other.

Example 2: Two pieces of puffed rice become equally charged as they are poured out ofthe box and into Kirk’s cereal bowl. If the force between the puffed rice piecesis 4 � 10�23 N when the pieces are 0.03 m apart, what is the charge on each ofthe pieces?

Solution: Because both charges are the same, solve for both q’s together. Thenfind the square root of that value to determine one of the charges.

Given: F � 4 � 10�23 N Unknown: q � ?d � 0.03 m Original equation: F �k � 9.0 � 109 N � m2/C2

Solve: q1q2 � � � 4 � 10�36 C2

This is the square of the charge on the pieces of puffed rice. To find the chargeon one piece of puffed rice, take the square root of this number.

q � � 2 � 10�18 C24 � 10�36 C2

14 � 10�23 N 2 10.03 m 229.0 � 109 N #m2>C2

Fd2

k

kq1q2

d2

19.0 � 109 N #m2>C2 2 12.0 � 10�6 C 2 12.0 � 10�6 C 210.70 m 22

kq1q2

d2

kq1q2

d2

198 Electrostatics

Practice Exercises

Exercise 1: When sugar is poured from the box into the sugar bowl, the rubbing of sugargrains creates a static electric charge that repels the grains, and causes sugarto go flying out in all directions. If each of two sugar grains acquires a chargeof 3.0 � 10�11 C at a separation of 8.0 � 10�5 m, with what force will theyrepel each other?

Answer:

Exercise 2: Boppo the clown carries two mylar balloons that rub against a circus elephant,causing the balloons to separate. Each balloon acquires 2.0 � 10�7 C of charge.How large is the electric force between them when they are separated by adistance of 0.50 m?

Answer:

Exercise 3: Inez uses hairspray on her hair each morning before going to school. Thespray spreads out before reaching her hair partly because of the electrostaticcharge on the hairspray droplets. If two drops of hairspray repel each otherwith a force of 9.0 � 10�9 N at a distance of 0.070 cm, what is the charge oneach of the equally-charged drops of hairspray?

Answer:

Electrostatics 199

219.0 � 10�9 N 2 17.0 � 10�4 m 22> 19.0 � 109 N #m2>C2 22Fd2>k

Exercise 4: Bonnie is dusting the house and raises a cloud of dust particles as she wipesacross a table. If two 4.0 � 10�14-C pieces of dust exert an electrostatic force of2.0 � 10�12 N on each other, how far apart are the dust particles at that time?

Answer:

Exercise 5: Each of two hot-air balloons acquires a charge of 3.0 � 10�5 C on its surfaceas it travels through the air. How far apart are the balloons if the electrostaticforce between them is 8.1 � 10�2 N?

Answer:

15-2 Electric FieldVocabulary Electric Field: An area of influence around a charged object. The magnitude of

the field is proportional to the amount of electrical force exerted on a positivetest charge placed at a given point in the field.

electric field � or E �

The SI unit of electric field is the newton per coulomb (N/C).

The electric field around a charged object is a vector and can be representedwith electric field lines that point in the direction of the force exerted on a unitof positive charge. In other words, electric field lines point away from apositive charge and toward a negative charge, as shown in the diagram.

Fqo

electric forcetest charge

200 Electrostatics

219.0 � 109 N #m2>C2 2 14.0 � 10�14 C 22> 12.0 � 10�12 N 22kq1q2>F

219.0 � 109 N #m2>C2 2 13.0 � 10�5 C 22> 18.1 � 10�2 N 22kq1q2>F

For a point charge (or other spherical charge distribution), the magnitude ofthe electric field can be written as

E � � �

where q is the charge on the surface of the object, and d is the distance betweenthe center of the charged object and a small positive test charge, qo, placed inthe field.

Solved Examples

Example 3: Deepika pulls her wool sweater over her head, which charges her body as thesweater rubs against her cotton shirt. What is the electric field at a locationwhere a 1.60 � 10�19 C-piece of lint experiences a force of 3.2 � 10�9 N as itfloats near Deepika? b) What will happen if Deepika now touches a conductorsuch as a door knob?

a. Given: qo � 1.60 � 10�19 C Unknown: E � ?F � 3.2 � 10�9 N Original equation: F � qoE

Solved: E � � � 2.0 � 1010 N/C

b. She will reduce her charge in a process called grounding, in which excesselectrons flow from her body into the ground and spread evenly over thesurface of Earth.

Example 4: A fly accumulates 3.0 � 10�10 C of positive charge as it flies through the air.What is the magnitude and direction of the electric field at a location 2.0 cmaway from the fly?


Given: k � 9.0 � 109 N � m2/C2 Unknown: E � ?q � 3.0 � 10�10 C Original equation: E �d � 0.020 m

Solve: E � � � 6800 N/C away fromthe fly

19.0 � 109 N #m2>C2 2 13.0 � 10�10 C 210.020 m 22

kq

d2

kq

d2

3.2 � 10�9 N1.60 � 10�19 C

Fqo

kq

d2

kqoq

qod2Fqo

Electrostatics 201

Practice Exercises

Exercise 6: Mr. Patel is photocopying lab sheets for his first period class. A particle oftoner carrying a charge of 4.0 � 10�9 C in the copying machine experiencesan electric field of 1.2 � 106 N/C as it’s pulled toward the paper. What is theelectric force acting on the toner particle?

Answer:

Exercise 7: As Courtney switches on the TV set to watch her favorite cartoon, the electronbeam in the TV tube is steered across the screen by the field between twocharged plates. If the electron experiences a force of 3.0 � 10�6 N, how largeis the field between the deflection plates?

Answer:

Exercise 8: Gordon the night custodian dusts off a classroom globe with a feather duster,causing the globe to acquire a charge of �8.0 � 10�9 C. What is the magnitudeand direction of the electric field at a point 0.40 m from the center of thecharged globe?

Answer:

202 Electrostatics

Exercise 9: April is decorating a tree in her backyard with plastic eggs in preparation forEaster. She hangs two eggs side by side so that their centers are 0.40 m apart.April rubs the eggs to shine them up, and in doing so places a charge on eachegg. The egg on the left acquires a charge of 6.0 � 10�6 C while the egg onthe right is charged with 4.0 � 10�6 C. What is the electric field at a point 0.15 m to the right of the egg on the left?

Answer:

15-3 Electrical Potential DifferenceVocabulary Potential Difference: The work done to move a positive test charge from one

location to another.

potential difference � or V �

The SI unit for potential difference is the volt (V), which equals a joule percoulomb (J/C).

Remember, the term “work” can be replaced with the term “energy,” becauseto store energy in, or give energy to, an object, work must be done. Therefore,potential difference can also be defined as the electrical potential energy perunit test charge. Voltage is often used to mean potential difference.

The field that exists between two charged parallel plates is uniform exceptnear the plate edges, and depends upon the potential difference between theplates and the plate separation.

electric field � or E �V¢d

potential difference

separation between plates

Wqo

worktest charge

Electrostatics 203

Here, the unit for electric field is the volt/meter. It was noted earlier that theunit for electric field is the newton/coulomb. This means that a volt/metermust equal a newton/coulomb.

� � �

Solved Examples

Example 5: An electron in Tammie’s TV is accelerated toward the screen across a potentialdifference of 22 000 V. How much kinetic energy does the electron lose whenit strikes the TV screen?

Given: qo � 1.60 � 10�19 C Unknown: W � ?V � 22 000 V Original equation: V �

Solve: W � qoV � (1.60 � 10�19 C)(22 000 V) � 3.5 � 10�15 J

Example 6: Amir shuffles his feet across the living room rug, building up a charge on his body. A spark will jump when there is a potential difference of 9000 Vbetween the door and the palm of Amir’s hand. This happens when his handis 0.3 cm from the door. At this point, what is the electric field between Amir’shand and the door?


Given: V � 9000 V Unknown: E � ?�d � 0.003 m Original equation: V � E�d

Solve: E � � � 3 � 106 V/m

Practice Exercises

Exercise 10: James recharges his dead 12.0-V car battery by sending 28 000 C of chargethrough the terminals. How much electrical potential energy must James storein the car battery to make it fully charged?

Answer:

9000 V0.003 m

V¢d

Wqo

newtoncoulomb

newton #metercoulomb #meter

joule>coulomb

metervolt

meter

204 Electrostatics

Exercise 11: If an electron loses 1.4 � 10�15 J of energy in traveling from the cathode to thescreen of Jeffrey’s personal computer, across what potential difference must ittravel?

Answer:

Exercise 12: A “bug zapper” kills bugs that inadvertently stray between the charged platesof the device. The bug causes sudden dielectric breakdown of the air betweenthe plates. If two plates in a bug zapper are separated by 5.0 cm and the fieldbetween them is a uniform 2.8 � 106 V/m, what is the potential differencethat kills the unsuspecting bugs?

Answer:

Exercise 13: While getting out of a car, Victor builds up a charge on his body as he slidesacross the cloth car seats. When he attempts to shut the car door, his handdischarges 12 000 V through a uniform electric field of 3.0 � 106 V/m. Howfar is his hand from the door at the time the spark jumps?

Answer:

Exercise 14: A lightning bolt from a cloud hits a tree after traveling 200 m to the groundthrough an electric field of 2.0 � 106 V/m. a) What is the potential differencebetween the cloud and the tree just before the lightning bolt strikes? b) If youare in an open field during a lightning storm and the only thing you seenearby is a tall tree, is it a good idea to stand under the tree for protectionfrom the lightning? Why or why not?

Answer: a.

Answer: b.

Electrostatics 205


A-1: A raindrop acquires a negative charge of 3.0 � 10�18 C as it falls. What is theforce of attraction when the raindrop is 6.0 cm from the bulb on the end of acar antenna that holds a charge of 2.0 � 10�6 C?

A-2: In a grain elevator on Farmer Judd’s farm, pieces of grain become electricallycharged while falling through the elevator. If one piece of grain is chargedwith 5.0 � 10�16 C while another holds 2.0 � 10�16 C of charge, what is theelectrostatic force between them when they are separated by 0.050 m?

A-3: Rocco, an auto body painter, applies paint to automobiles by electricallycharging the car’s outer surface and oppositely charging the paint particlesthat he sprays onto the car. This causes the paint to adhere tightly to the car’s surface. If two paint particles of equal charge experience a force of4.0 � 10�8 N between them at a separation of 0.020 cm, what is the charge on each?

A-4: After unpacking a shipment of laboratory glasswear, Mrs. Payne dumps thebox of Styrofoam packing chips into a recycling bin. The chips rub togetherand two chips 0.015 m apart repel each other with a force of 6.0 � 10�3 N.What is the charge on each of the chips?

A-5: Wiz the cat is batting at two Ping-Pong balls hanging from insulating threads with their sides just barely touching. Each ball acquires a positivecharge of 3.5 � 10�9 C from Wiz’s fur and they swing apart. a) If a force of6.0 � 10�5 N acts on one of the balls, how far apart are they from each other?b) Is the force between them one of attraction or repulsion?

A-6: A droplet of ink in an ink-jet printer carrying a charge of 8.0 � 10�13 C isdeflected onto the paper by a force of 3.2 � 10�4 N. How strong is the fieldthat causes this force?

A-7: In the human body, nerve cells work by pumping sodium ions out of a cell inorder to maintain a potential difference across the cell wall. If a sodium ioncarries a charge of 1.60 � 10�19 C as it is pumped with an electrical force of2.0 � 10�12 N, what is the electric field between the inside and outside of thenerve cell?

A-8: Each of two Van de Graaff generators, whose centers are separated from oneanother by 0.50 m, becomes charged after they are switched on. One Van deGraaff generator holds �3.0 � 10�2 C while the other holds �2.0 � 10�2 C.What is the magnitude and direction of the electric field halfway betweenthem?

A-9: Willa the witch dusts her crystal ball with her silk scarf, causing the ball tobecome charged with 5.0 � 10�9 C. Willa then stares into the crystal ball andthe wart on the end of her nose experiences an electric field strength of 2200 N/C. How far is the tip of her nose from the center of the crystal ball?

206 Electrostatics

A-10: The Millikan oil drop experiment of 1909 allowed Robert A. Millikan todetermine the charge of an electron. In the experiment, an oil drop issuspended between two charged plates by an electric force that equals thegravitational force acting on the 1.1 � 10�14-kg drop. a) What is the charge onthe drop if it remains stationary in an electric field of 1.72 � 105 N/C? b) Howmany extra electrons are there on this particular oil drop?

A-11: In eighteenth-century Europe, it was common practice to ring the church bellsin an attempt to ward off lightning. However, during one 33-year period,nearly 400 church steeples were struck while the bells were being rung. If abolt of lightning discharges 30.0 C of charge from a cloud to a steeple across apotential difference of 15 000 V, how much energy is lost by the cloud andgained by the steeple?

A-12: In Exercise A-7, how thick is the wall of the nerve cell if there is a potentialdifference of 0.089 between the inside and outside of the cell?

A-13: Ulrich stands next to the Van de Graaff generator and gets a shock as he holdshis knuckle 0.2 m from the machine. In order for a spark to jump, the electricfield strength must be 3 � 106 V/m. At this distance, what is the potentialdifference between Ulrich and the generator?


B-1: Three glass Christmas balls become electrically charged when Noel removes them from thepackaging material in their box. Noel hangs theballs on the tree as shown. If each ornamenthas acquired a charge of 2.0 � 10�10 C, what isthe magnitude and direction of the forceexperienced by the ball at the top?

B-2: In a TV picture tube, electrons are accelerated from rest up to very highspeeds through a potential difference of 22 000 V. At what speed will anelectron be moving just as it strikes the TV screen? (In reality you would haveto consider the effects of relativity in order to solve this exercise properly;however, ignore such relativistic effects here.)

B-3: A lightning bolt discharges into New Hampshire’s Lake Winnipesaukee afterpassing through a potential difference of 9.00 � 107 V. What is the minimumamount of charge the lightning bolt could be carrying, if it were to vaporize1000. kg of water in the lake that was originally at a temperature of 20.0 °C?

Electrostatics 207

16 Direct Current Circuits

209

16-1 Current and ResistanceVocabulary Current: The amount of charge that passes through an area in a given amount

of time.

current � or I �

The SI unit for current is the ampere (A), which equals one coulomb persecond (C/s).

In conductors, such as metal wires, electrons are relatively free to move, andcan carry energy throughout a circuit. This energy comes from a source suchas a battery that converts chemical energy into electrical energy for use in thecircuit. As energy is transformed in a battery, a potential difference, V,develops across the battery’s terminals. This potential difference is called anelectromotive force, or EMF. In this book, voltage between the terminals of abattery is simply referred to as potential difference.

Vocabulary Resistance: An opposition to the flow of charge.

For a given source voltage, the resistance of a circuit determines how muchcharge will flow in the circuit. When charge passes through a resistance, someelectrical energy is changed to other forms. This is produced by a potentialdifference across the resistance.

potential difference � (current)(resistance) or V � IR

The SI unit for resistance is the ohm (�), which equals one volt per amp (V/A).

Sometimes it is not desirable to use wires that have a high resistance, becauseconsiderable energy losses occur when charge flows through a resistor.However, in any device that produces heat, such as a toaster, high resistance isneeded or else the toaster would not get hot. Therefore, a heating elementmade with superconducting wires would be useless.

The resistance of a wire depends upon the type of material that the wire ismade of, its length, and its cross-sectional area. The longer the wire, the moreresistant it is to the flow of charge. The larger the cross-sectional area of thewire, the less resistant it is to charge flow. Temperature also affects the

¢q

¢t

charge

time

resistance of a wire. The hotter the wire, the more resistant it becomes to theflow of charge. This means that more current will flow through a toaster whenit is first turned on than when the coils are glowing red hot.

Solved Examples

Example 1: Household current in a circuit cannot generally exceed 15 A for safety reasons.What is the maximum amount of charge that could flow through this circuitin a house during the course of a 24.0-h day?

Solution: Because the unit ampere means coulombs per second, 24.0 h mustbe converted in 86 400 s.

Given: I � 15 A Unknown: �q � ?�t � 86 400 s Original equation: I �

Solve: �q � I�t � (15 A)(86 400 s) � 1.3 � 106 C

Example 2: What is the resistance of the heating element in a car lock de-icer that containsa 1.5-V battery supplying a current of 0.5 A to the circuit?

Given: V � 1.5 V Unknown: R � ?I � 0.5 A Original equation: V � IR

Solve: R � � � 3 �

Practice Exercises

Exercise 1: Arthur is going trick-or-treating for Halloween so he puts new batteries in hisflashlight before leaving the house. Until the batteries die, it draws 0.500 A ofcurrent, allowing a total of 5400. C of charge to flow through the circuit. Howlong will Arthur be able to use the flashlight before the batteries’ energy isdepleted?

Answer:

1.5 V0.5 A

VI

¢q

¢t


Exercise 2: Fabian’s car radio will run from the 12-V car battery that produces a currentof 0.20 A even when the car is turned off. The car battery will no longeroperate when it has lost 1.2 � 106 J of energy. If Fabian gets out of the car andleaves the radio on by mistake, how long will it take for the car battery to gocompletely dead (that is, lose all energy)?

Answer:

Exercise 3: While cooking dinner, Dinah’s oven uses a 220.-V line and draws 8.00 A ofcurrent when heated to its maximum temperature. What is the resistance ofthe oven when it is fully heated?

Answer:

Exercise 4: Justine’s hair dryer has a resistance of 9.00 � when first turned on. a) Howmuch current does the hair dryer draw from the 110.-V line in Justine’s house?b) What happens to the resistance of the hair dryer as it runs for a long time?

Answer: a.

Answer: b.

Exercise 5: Camille takes her pocket calculator out of her bookbag as she gets ready to doher physics homework. In the calculator, a 0.160-C charge encounters 19.0 � ofresistance every 2.00 seconds. What is the potential difference of the battery?

Answer:

Direct Current Circuits 211

16-2 CapacitanceVocabulary Capacitor: A device that stores charge on conductors that are separated by

an insulator.

Capacitance is a measure of the amount of charge stored on the conductors,for a given potential difference.

capacitance � or C �

The SI unit for capacitance is the farad (F), which equals one coulomb pervolt (C/V).

A capacitor may be used in a circuit by storing charge on two parallel platesand then periodically releasing it into the circuit, creating an intermittent flowof charge.

Solved Examples

Example 3: The first capacitor was invented by Pieter van Musschenbroek in 1745 when heand his assistant stored charge in a device called a Leyden jar. If 5 � 10�4 C ofcharge were stored in the jar over a potential difference of 10 000 V, what wasthe capacitance of the Leyden jar? (When van Musschenbroek touched the jar,he received such a large jolt that he exclaimed he would not try the experimentagain for all the kingdom of France!)

Given: �q � 5 � 10�4 C Unknown: C � ?V � 10 000 V Original equation: C �

Solve: C � � � 5 � 10�8 F

Example 4: Lydia pushes the shutter button of her camera and the flash unit releases the4.5 � 10�3 C of charge that was stored in a 500.-�F capacitor. What is thepotential difference across the plates of the capacitor inside the flash?

Solution: The term � (micro) means 10�6, so a �F means 10�6 farad.

Given: �q � 4.5 � 10�3 C Unknown: V � ?C � 500. � 10�6 F Original equation: C �

Solve: V � � � 9.0 V4.5 � 10�3 C500. � 10�6 F

¢q

C

¢q

V

5 � 10�4 C10 000 V

¢q

V

¢q

V

¢q

V

amount of charge

potential difference


Practice Exercises

Exercise 6: The nervous system of the human body contains axons whose membranes actas small capacitors. A membrane is capable of storing 1.2 � 10�9 C of chargeacross a potential difference of 0.070 V before discharging nerve impulsesthrough the body. What is the capacitance of one of these axon membranes?

Answer:

Exercise 7: During a lightning storm, the separation between the clouds and the earthacts as a giant capacitor with a capacitance of 2500 �F. If the transmittingtower of radio station KBOZ is hit by a bolt of lightning carrying 50. C ofcharge, what is the potential difference between the cloud and the tower?

Answer:

Exercise 8: Dr. Frankenstein brings his monster to life with electroshock treatment bydischarging a 50.-�F capacitor through the monster’s neck across a potentialdifference of 24 V. How much charge flows into the monster to make himcome alive?

Answer:

Exercise 9: On Saturday nights, Greg likes to go the Frisco Disco, where he can danceunder the strobe light. The strobe contains a 200-�F capacitor that storescharge over a 1000-V potential difference. If the strobe flashes 4 times eachsecond, what is the current flow created by the strobe’s capacitor?

Answer:


16-3 PowerVocabulary Power: The amount of work done in a given unit of time.

As seen in the previous chapter, electrical work is done when an amount ofcharge, �q, is transferred across a potential difference, V, or W � �qV. Thefaster this transfer of charge occurs, the more power is generated in the circuit.

Power � or P � � � IV

Therefore, as current is drawn in a circuit to power an appliance, a potentialdifference occurs across the appliance.

The SI unit for electrical power is the watt (W), which equals one joule persecond (J/s).

Solved Examples

Example 5: The lighter in Bryce’s car has a resistance of 4.0 �. a) How much current doesthe lighter draw when it is run off the car’s 12-V battery? b) How muchpower does the lighter use?

a. Given: R � 4.0 � Unknown: I � ?V � 12 V Original equation: V � IR

Solve: I � � � 3.0 A

b. Given: I � 3.0 A Unknown: P � ?V � 12 V Original equation: P � IV

Solve: P � IV � (3.0 A)(12 V) � 36 W

Example 6: A 120.-V outlet in Carol’s college dorm room is wired with a circuit breaker ona 5-A line so that students cannot overload the circuit. a) If Carol tries to iron ablouse for class with her 700-W iron, will she trip the circuit breaker? b) Whatis the resistance of the iron?

Solution: A circuit breaker is a switch that automatically turns a circuit off ifthe current is too high.

a. Given: P � 700. W Unknown: I � ?V � 120. V Original equation: P � IV

Solve: I � � � 5.83 A Yes, she will!700. W120. V

PV

12 V4.0 �

VR

¢qV

¢tW¢t

workelapsed time


It may be difficult to see how a watt/volt equals an amp until you begin tobreak down the units.

� � � amp

b. Now find the resistance using V � IR.

Given: V � 120. V Unknown: R � ?I � 5.83 A Original equation: V � IR

Solve: R � � � 20.5 �

Example 7: The Garcias like to keep their 40.0-W front porch light on at night to welcomevisitors. If the light is on from 6 p.m. until 7 a.m., and the Garcias pay 8.00¢per kWh, how much does it cost to run the light for this amount of time each week?

Solution: First, convert the power units to kilowatts, kW, because the cost ofhousehold energy is measured in kWh. 40.0 W � 0.0400 kW

Next, determine how long the light is left on each week. From 6 P.M. until 7 A.M. is 13 h. Operating 7 days a week means that the light is on for a total of91.0 hours.

Given: P � 0.0400 kW Unknown: W � ? Cost � ?�t � 91.0 h

Original equation: P �

Solve: W � P�t � (0.0400 kW)(91.0 h) � 3.64 kWh

Cost � (3.64 kWh) � 29.1¢

Therefore, it costs the Garcias about 29¢ to run the light all night for an entireweek, or a little over $15 per year.

Practice Exercises

Exercise 10: How much power is used by a contact lens heating unit that draws 0.070 A ofcurrent from a 120-V line?

Answer:

8.00¢1.00 kWh

W¢t

120. V5.83 A

VI

coulombsecond

joule>second

joule>coulombwattvolt


Exercise 11: Celeste’s air conditioner uses 2160 W of power as a current of 9.0 A passesthrough it. a) What is the voltage drop when the air conditioner is running? b) How does this compare to the usual household voltage? c) What wouldhappen if Celeste tried connecting her air conditioner to a usual 120-V line?

Answer: a.

Answer: b.

Answer: c.

Exercise 12: Which has more resistance when plugged into a 120.-V line, a 1400.-Wmicrowave oven or a 150.-W electric can opener?

Answer:

Exercise 13: Valerie’s 180-W electric rollers are plugged into a 120-V line in her bedroom.a) What current do the electric rollers draw? b) What is the resistance of therollers when they are heated? c) Combining the equations just used, derive anequation that relates power to voltage and resistance.

Answer: a.

Answer: b.

Answer: c.


Exercise 14: Mrs. Olsen leaves her 0.900-kW electric coffee maker on each day as she headsoff to work at 6 A.M. because she likes to come home to a hot cup of coffee at 6 P.M. a) If the electric company charges Mrs. Olsen $0.100 per kWh, howmuch does running the coffee maker cost her each day? b) What is the yearlycost to run the coffee maker?

Answer: a.

Answer: b.

Exercise 15: While writing this book, the author spent about 1000 h working on herpersonal computer that has a power input of 60.0 W. Seventy additional hourswere spent with the 60.0-W computer and the 240.-W printer running. Howmuch did it cost for the energy use of these two devices, at a cost of $0.100 perkWh?

Answer:

16.4 Series and Parallel CircuitsWhen multiple resistors are used in a circuit, the total resistance in the circuitmust be found before finding the current. Resistors can be combined in acircuit in series or in parallel.

Resistors in Series

When connected in series, the total resistance, RT, is equal to

RT � R1 � R2 � R3 � . . .

In series, the total resistance is always larger than any individual resistance.


Current in series resistors: In series circuits, charge has only one path throughwhich to flow. Therefore, the current passing through each resistor in series isthe same.

Potential difference across series resistors: As charge passes through each ofthe resistors, it loses some energy. This means that there will be a potentialdifference across each resistor. The sum of all the potential differences equalsthe potential difference across the battery, assuming negligible resistance inthe connecting wires.

Resistors in Parallel

When connected in parallel, the total resistance, RT, is equal to

� � � � . . .

Don’t forget! After finding a commondenominator and determining the sum of thesefractions, flip over the answer to determine RT.

In parallel circuits, the total resistance is always smaller than any individualresistance.

Current in parallel resistors: In parallel circuits, there is more than onepossible path and current divides itself according to the resistance of eachpath. Since current will take the “path of least resistance,” the smallest resistorwill allow the most current through, while the largest resistor will allow theleast current through. The sum of the currents in each parallel resistor equalsthe original current entering the branches.

Potential difference in parallel resistors: The potential difference across eachof the resistors in a parallel combination is the same. If there are no otherresistors in the circuit, it is equal to the potential difference across the battery,assuming negligible resistance in the connecting wires.

Solved Examples

Example 8: Find the total resistance of the three resistors connected in series.

Solve: RT � R1 � R2 � R3 � 12 � � 4 � � 6 � � 22 �

1R3

1R2

1R1

1RT


Example 9: Find the total resistance of the same three resistors now connected in parallel.

Solve: � � � � � �

� � � � � RT � 2 �

Example 10: Find the total current passing through the circuit.

This circuit contains resistors in parallel that are then combined with a resistor in series. Alwaysbegin solving such a resistor combination byworking from the inside out. In other words, firstdetermine the equivalent resistance of the tworesistors in parallel before combining this totalresistance with the one in series.

Look first at the parallel combination.

� � � � � � � �

RT � 4 �

Now, combine this equivalent resistance with the resistor in series.

RT � R1 � R2 � 4 � � 4 � � 8 �

To find the current flowing through the circuit, use this total resistance incombination with the potential difference from the battery.

Given: V � 16 V Unknown: I � ?R � 8 � Original equation: V � IR

Solve: I � � � 2 A

Example 11: Find the current in the 9-� resistor.

For the parallel branch

� � � � � � � �

RT � 6 �

Combining with the series resistor

RT � R1 � R2 � 6 � � 6 � � 12 �

16 �

318 �

118 �

218 �

118 �

19 �

1R2

1R1

1RT

16 V8 �

VR

14 �

312 �

212 �

112 �

16 �

112 �

1R2

1R1

1RT

12 �

612 �

212 �

312 �

112 �

1RT

16 �

14 �

112 �

1R3

1R2

1R1

1RT



Solve: I � � � 3 A

This 3 A is the current through the entire circuit. Use this current to find thepotential difference across the parallel combination. Remember, the potentialdifference across resistors wired in parallel is the same regardless of whichpath is taken. Because the resistors in parallel have a combined resistance of 6 �, you find the potential difference across the parallel branch as follows.

Given: R � 6 � Unknown: V � ?I � 3 A Original equation: V � IR

Solve: V � IR � (3 A)(6 �) � 18 V

Therefore, the potential difference across both the top and the bottom branchesis 18 V. Now use this 18-V drop to determine the current in the 9-� resistor.


Solve: I � � � 2 A

Practice Exercises

Exercise 16: Using the diagram, a) find the total resistance in the circuit. b) Find the totalcurrent through the circuit.

Answer: a.

Answer: b.


Answer: a.

Answer: b.

18 V9 �

VR

36 V12 �

VR



Answer: a.

Answer: b.

Exercise 19: Old-fashioned holiday lights were connected in series across a 120-Vhousehold line. a) If a string of these lights consists of 12 bulbs, what is thepotential difference across each bulb? b) If the bulbs were connected inparallel, what would be the potential difference across each bulb?

Answer: a.

Answer: b.

Exercise 20: Before going to work each morning, Gene runs his 18-� toaster, 11-� electricfrying pan, and 14-� electric coffee maker, all at the same time. The three areconnected in parallel across a 120-V line. a) What is the current through eachappliance? b) If a household circuit could carry a maximum current of 15 A,would Gene be able to run all of these appliances at the same time?

Answer: a.

Answer: b.


Exercise 21: Timmy is playing with a new electronics kit he has received for his birthday.He takes out four resistors with resistances of 15 �, 20 �, 20 �, and 30 �.a) How would Timmy have to wire the resistors so that they would allow themaximum amount of current to be drawn? Calculate the total resistance inthis circuit. b) How must he wire the resistors so that they draw a minimumamount of current? Calculate the total resistance in this circuit.

Answer: a.

Answer: b.

Exercise 22: Farmer Crockett is preparing tomato seedlings for his spring planting bygrowing the small plants over five 46-� strip heaters wired in parallel. a) Howmuch current does each heater draw from a 120-V line? b) How much currentdo they draw all together?

Answer: a.

Answer: b.


A-1: Otto accidently leaves his automobile headlights on overnight and is unableto start his car in the morning. Each of the two headlights connected inparallel draws 2.00 A of current from the 12.0-V battery. If the battery stores7.50 � 105 J of energy, how long will it take for the headlights to go off? b) Why are the headlights connected in parallel?

A-2: Officer Moynihan is patrolling his beat with a 4.5-V flashlight whose lightbulbhas a resistance of 12 �. How much current does the flashlight draw?

A-3: Each night before falling asleep, Linus turns on his electric blanket that isplugged into the 120.-V electrical outlet. A current of 1.20 A flows through theblanket. a) What is the blanket’s resistance? b) Does Linus want his electricblanket to have a high resistance or a low resistance? Why?


A-4: Herbert had just suffered a heart attack but he was revived in the hospitalemergency room with a device called a defibrillator. (The paddles of adefibrillator supply a short pulse of high voltage to restart the heart.) Thedefibrillator contains a 20.-�F capacitor that releases 0.15 C of charge. a) Whatis the potential difference between the defribrillator paddles during thedischarge? b) Why do you think doctors yell “Clear!” to the attendants beforedischarging the defibrillator?

A-5: Sherm is typing his term paper on a computer that contains a high-speedswitch, controlled with a small 100 � 10�12 F speed-up capacitor. What is thecurrent flow created by the capacitor if it discharges every 0.1 s across apotential difference of 5 V?

A-6: Every Sunday morning Stuart makes “breakfast in bed” for his wife. However,because the household wires can only carry a maximum current of 15 A fromthe 120.-V line, it is difficult to run all of the appliances simultaneously withoutblowing a fuse. What is the most power Stuart may use while cooking, beforeblowing a fuse?

A-7: In the previous exercise, a) how much current will Stuart draw if he tries torun the 700.-W toaster and 1000.-W coffee maker at the same time? b) Will thiscause him to blow the fuse?

A-8: Xiaoyi’s aquarium operates for 24.0 h a day and contains a 5.0-W heater, two20.0-W lightbulbs, and a 35.0-W electric filter. If Xiaoyi pays $0.100 per kWhfor her electricity bill, how much will it cost to maintain the aquarium for 30.0 days?

A-9: The average power plant, running at full capacity, puts out 500. MW of power.If the power company charges its customers $0.10 per kWh, what is therevenue brought in by the power plant each day?

A-10: Horace has invented a unique pair of reading glasses that have two smalllight bulbs at the bottom wired in series, so that he can see the newspaperwhen he is reading at night. Each of the bulbs has a resistance of 2.00 �, andthe system runs off a 3.20-V battery. How much current is drawn by Horace’sreading glasses?

A-11: Jay has two 8-� stereo speakers wired in series in the front of his car connectedto the 4.0-V output of the stereo. a) What is the current through each of thespeakers? b) In his garage, Jay finds two more old speakers with resistances of4 � and 16 �. He wires each in parallel with the 8-� combination. What is thenew current through the 8-� speakers? c) If the loudness of each speaker isproportional to the amount of power used, how has the loudness of the two 8-� speakers changed?


A-12: Find a) the total resistance in circuit A below. b) Find the total current throughthe circuit.

A-13: Find a) the total resistance in circuit B below. b) Find the total current throughthe circuit.

A-14: Find a) the total resistance in circuit C below. b) Find the total current throughthe circuit.

Circuit A Circuit B Circuit C


B-1: An 800.-W submersible electric heater is put into a 20.0 °C hottub until the50.0-kg of tub water has warmed up to 70.0 °C. How long will it take for theheater to heat the tub water? (cwater � 4187 J/kg°C)

B-2: Find the total current in the circuit in the diagram.

B-3: In exercise A-10, the light bulbs are rated for 5 h of use before they burn out. If the battery can supply 5184 J to the circuit, which occurs first, energydepletion in the battery or failure of a bulb?


225

17-1 Magnetic Forces and FieldsVocabulary Magnetic Field: An area of influence around a moving charge. The size of the

field is related to the amount of magnetic force experienced by the movingcharge when it is at a given location in the field.

magnetic field � or B �

The SI unit for magnetic field is the tesla (T), which equals one netwon peramp �meter (N/A�m).

When solving for the magnetic force, rewrite this equation as F � qvB.

The magnitude of the magnetic force can also be written in terms of thecurrent, I, flowing through a length of wire, L.

force � (current)(length of wire)(magnetic field) or F � ILB

Unlike gravitational force or electric force, magnetic force is perpendicular tothe plane formed by the field and the moving charge, and is greatest when themagnetic field and the current are perpendicular to each other.

The easiest way to detect a magnetic field is with a compass.

Solved Examples

Example 1: A proton speeding through a synchrotron at 3.0 � 107 m/s experiences amagnetic field of 4.0 T that is produced by the steering magnets inside thesynchrotron. What is the magnetic force pulling on the proton?

Solution: Remember, the charge of a proton or an electron is 1.60 � 10�19 C.

Given: q � 1.60 � 10�19 C Unknown: F � ?v � 3.0 � 107 m/s Original equation: F � qvBB � 4.0 T

Solve: F � qvB � (1.60 � 10�19 C)(3.0 � 107 m/s)(4.0 T) � 1.9 � 10�11 N

Fqv

force(charge)(speed)

17 Magnetism and Electromagnetic Induction

Example 2: A 10.0-m-long high-tension power line carries a current of 20.0 Aperpendicular to Earth’s magnetic field of 5.5 � 10�5 T. What is the magneticforce experienced by the power line?

Given: I � 20.0 A Unknown: F � ?L � 10.0 m Original equation: F � ILBB � 5.5 � 10�5 T

Solve: F � ILB � (20.0 A)(10.0 m)(5.5 � 10�5 T) � 0.011 N

Practice Exercises

Exercise 1: Dean is hunting in the Northwest Territories at a location where Earth’smagnetic field is 7.0 � 10�5 T. He shoots by mistake at a duck decoy, and therubber bullet he is using acquires a charge of 2.0 � 10�12 C as it leaves hisgun at 300. m/s, perpendicular to Earth’s magnetic field. What is themagnitude of the magnetic force acting on the bullet?

Answer:

Exercise 2: A wasp accumulates 1.0 � 10�12 C of charge while flying perpendicular toEarth’s magnetic field of 5.0 � 10�5 T. How fast is the wasp flying if themagnetic force acting on it is 6.0 � 10�16 N?

Answer:

Exercise 3: Kron, the alien freedom fighter from the planet Krimbar, shoots his gun thatfires protons at a speed of 3.0 � 106 m/s. a) What is Krimbar’s magnetic fieldif it creates a force of 2.88 � 10�15 N on the protons? b) How does thiscompare to Earth’s magnetic field?

Answer: a.

Answer: b.


Exercise 4: The magnetic field in Boston, Massachusetts has a horizontal component tothe north of 0.18 � 10�4 T and a vertical component of 0.52 � 10�4 T straightdownward. a) What is the magnitude and direction of Earth’s magnetic fieldin Boston? b) If a 2.0-m-long household wire is carrying a current of 15 A in adirection perpendicular to the field, what is the magnitude of the magneticforce experienced by the wire?

Answer: a.

Answer: b.

17-2 Electromagnetic Induction

Magnetic Flux and Induced Voltage

Vocabulary Flux: The number of magnetic field lines passing through a given area.

flux � (area)(perpendicular component of the magnetic field)

or � AB

The unit for flux is the weber (Wb), which equals one tesla�meter squared(T �m2).

Therefore, if a loop of wire lies perpendicular to a magnetic field, themaximum possible number of lines of flux will pass through the loop. If theloop of wire lies parallel to the field, the flux through the loop will be zero.

Vocabulary Faraday’s Law: If the flux through a given area changes over time, a voltagewill be induced in the wire and a current will momentarily flow. If thenumber of turns of wire is increased, the voltage will increase proportionally.

potential difference �

or V �

Note: This potential difference is also referred to as the induced voltage.

N¢

¢t

(number of turns)(change in flux)elapsed time

Magnetism and Electromagnetic Induction 227

210.18 � 10�4 T 22 � 10.52 � 10�4 T 22

Vocabulary Lenz’s Law: An induced voltage always produces a magnetic field thatopposes the field that originally produced it.

In other words, if the original magnetic field, and thus the flux, is goingtoward the north, the induced voltage will produce an opposing field and flux that goes toward the south.

Transformers

Vocabulary Transformer: A device that produces a change in voltage in an alternatingcurrent circuit.

A transformer consists of an iron core wound with a primary coil and a secondary coil. An alternatingcurrent placed through the primary coil induces achanging magnetic field through the core, which, inturn, induces a voltage in the secondary coil.

� or �

If the primary coil has more turns than the secondary coil, the transformerwill step down, or decrease, the incoming voltage. If the primary coil hasfewer turns than the secondary coil, the transformer will step up, or increase,the incoming voltage.

Solved Examples

Example 3: Tyrone is pedaling his bike down the street perpendicular to Earth’s magnetic field of 5.5 � 10�5 T. What is the flux through the metal rim of his bicyclewheel, if the wheel has an area of 1.13 m2?

Given: A � 1.13 m2 Unknown: � ?B � 5.5 � 10�5 T Original equation: � AB

Solve: � AB � (1.13 m2)(5.5 � 10�5 T) � 6.2 � 10�5 Wb

Example 4: If the bicycle in Example 3 takes 2.0 s to make a 90° turn onto a northboundstreet, what is the induced voltage in one metal rim of the bicycle?

Given: N � 1 turn Unknown: V � ? � 6.2 � 10�5 Wb Original equation: V �

�t � 2.0 s

Solve: V � � � 3.1 � 10�5 V11 turn 2 16.2 � 10�5 Wb 2

2.0 sN¢

¢t

N¢

¢t

Np

Ns

Vp

Vs

turns in primary coil

turns in secondary coil

voltage in primary coil

voltage in secondary coil


Example 5: While out for a walk with his mother, Lance notices a large, cylindrical graybox high atop a telephone pole. His mother explains that it is a transformer.This transformer takes 6000. V from the power company and steps it down tothe 240 V supplied to each of the houses on the street, with the use of asecondary coil containing 100. turns. How many turns are there in theprimary coil?

Given: Vp � 6000. V Unknown: Np � ?

Vs � 240 V Original equation: �

Ns � 100. turns

Solve: Np � � � 2500 turns

Practice Exercises

Exercise 5: Patty is driving down the expressway on her way to the office in a townwhere the horizontal component of Earth’s magnetic field is 3.5 � 10�5 T tothe north. The driver’s side window of Patty’s car has an area of 0.40 m2.a) What is the magnitude of the flux through the window if the car is movingsouth? b) How does it differ if the car is moving west?

Answer: a.

Answer: b.

Exercise 6: A medical process called magnetic resonance imaging (MRI) replaces X-rays insome instances where pictures are required to study internal organs. Eleanoris undergoing an MRI procedure and is placed inside a chamber housing thecoil of a large electromagnet that has a radius of 25.0 cm. A flux of 0.290 Wbpasses through the coil opening. What is the magnetic field inside the coil?

Answer:

16000. V 2 1100. turns 2240 V

VpNs

Vs

Np

Ns

Vp

Vs


Exercise 7: The hood ornament on Abe’s sedan is shaped like a ring 8.00 cm in diameter.Abe is driving toward the west so that Earth’s 5.00 � 10�5 T field provides noflux through the hood ornament. What is the induced voltage in the metalring as Abe turns from this street onto one where he is traveling north, if hetakes 3.0 s to make the turn?

Answer:

Exercise 8: Becky wears glasses whose wire frames are shaped like two circles, each withan area of 2.0 � 10�3 m2. The horizontal component of Earth’s magnetic fieldin Becky’s hometown is 1.9 � 10�5 T. If Becky turns her head back and forth,rotating it through 90° every 0.50 s, what is the induced voltage in the wireframe of one eyepiece?

Answer:

Exercise 9: Audrey disassembles the control box of her electric train and finds a smalltransformer inside. Its primary coil is made up of 600. turns and thesecondary coil is made up of 60. turns. a) If the household voltage supplied tothe train is 120 V, what voltage is required to make the train run? b) Is this astep-up or a step-down transformer?

Answer: a.

Answer: b.

Exercise 10: A hydroelectric plant in Niagara Falls sends 3000 V to the transformer in asubstation that steps it up to 120 000 V for transmission to homes in New YorkCity. If the primary coil contains 2000 turns, how many turns are there in thesecondary coil of the step-up transformer?

Answer:



A-1: In the giant CERN particle accelerator in Switzerland, protons are acceleratedto speeds of 2.0 � 108 m/s through a magnetic field of 3.5 T and then collidedwith a fixed target. What is the magnitude of the magnetic force experiencedby the protons as they are accelerated around the giant ring?

A-2: In Fred’s color TV, electrons are shot toward the screen through a 1.0 � 10�3-Tmagnetic field set up in the picture tube. a) If each electron experiences a magnetic force of 2.9 � 10�15 N, at what speed is it propelled through thepicture tube? b) How does this speed compare to the speed of light?

A-3: A proton shot out of the sun at a speed of 6.0 � 106 m/s during a “sunspotmaximum” travels through the strong magnetic field of the sun. What is themaximum magnetic force experienced by the proton at a point where thesun’s magnetic field is 0.090 T?

A-4: A 0.90-m-long straight wire on board the Voyager spacecraft carries a currentof 0.10 A perpendicular to Jupiter’s strong magnetic field of 5.0 � 10�4 T.What is the magnitude of the magnetic force experienced by the wire?

A-5: While vacuuming the living room rug, Buster pulls the 4.0-m vacuum cleanercord so that it is lying perpendicular to Earth’s magnetic field of 5.3 � 10�5 T.a) If the cord is carrying a current of 6.0 A, how large a magnetic force iscreated on the cord by Earth’s magnetic field? b) If Buster then pulls the cordso that it lies parallel to Earth’s magnetic field, how large is the magnetic forcenow experienced by the cord?

A-6: At the equator, where Earth’s 3.0 � 10�5-T magnetic field is parallel to thesurface of Earth, Donna is spinning her wedding ring (which has a diameterof 2.0 cm) on top of the table. Find the change in flux through the ring ifDonna a) slides it horizontally across the table, b) rolls it across the table, c) spins it on its edge.

A-7: Amanda’s little brother spins a bar magnet whose magnetic field is 3.0 � 10�2 T over the face of Amanda’s electric watch, perpendicular to acircular loop of wire of radius 0.60 cm inside the watch. a) What is theinduced voltage in the wire if the magnet is spun over the watch in 0.30 s? b) Why is it a bad idea to put an electric watch too close to a strong magnetic field?

A-8: While Hiroshi sits in his living room, the newspaper carrier rings his doorbell.If a voltage of 120 V passes through the 200-turn primary coil of thetransformer, how many turns are needed in the secondary coil to reduce thevoltage to the 6.0 V needed to run the doorbell?

A-9: A bug zapper in the Snyders’ back yard runs off a 120-V household linethrough a transformer whose primary coil contains 50. turns while thesecondary coil contains 2000. turns. a) What is the output voltage of thetransformer? b) Is this a step-up or a step-down transformer?



B-1: When Helen turns on the TV set, electrons are accelerated through a 20 000.-Vpotential difference and deflected by a 1.0 � 10�2-T magnetic field. What isthe average magnetic force experienced by an electron? (me � 9.11 � 10�31 kg)

B-2: Captain Kittredge is sailing due north, as indicated by his compass needle, in a location where Earth’s magnetic field is2.0 � 10�5 T. The captain inadvertently places his radio nearthe compass, allowing the wire from his radio to align in anorth-south direction. The 0.80-m-long wire carries a currentof 5.0 A and produces a magnetic force on the compassneedle of 2.8 � 10�4 N. To what angle will the compass needle turn while thewire is over it?

B-3: A velocity selector is a device that measures the speed of a charged particle byshooting the particle through oppositely charged plates enclosed in a tube.Inside the tube is a constant magnetic field, B. If a particle is to travel,undeflected, down the center of the tube, the magnetic force must equal theelectric force. If the magnetic field of 0.630 T is perpendicular to the electricfield of 5.00 � 104 N/C, find the speed of an electron sent through thevelocity selector.

B-4: An alpha particle (He nucleus) is shot at 5.0 � 106 m/s into a magnetic fieldof 0.20 T in a device known as a mass spectrometer. What is the radius of thepath followed by the alpha particle? (Hint: He nuclei contain 2 protons and 2 neutrons, each with a mass of 1.67 � 10�27 kg.)


18 Modern Physics

233

18-1 The Atom and the Quantum

Photon Energy

Vocabulary Quantum: A packet of energy that exhibits both particle and wave properties.

A quantum of light energy is called a photon. The photon’s energy is directlyproportional to the frequency of its lights. This can be written as

energy � (Planck’s constant)(frequency) or E � hf

where Planck’s constant, h, is equal to 6.63 � 10�34 J �s.

Recall from Chapter 13 that c � �f. So, the frequency of light can be written as

f � . Therefore, the energy of a photon is

energy � or E �

The common unit for the wavelength of light is the nanometer (nm), whichequals 10�9 m.

It is important to note that because the energy of a photon is so small,scientists rarely use the unit “joule” when describing this energy. Instead, asmaller unit, the electron volt (eV) is more commonly used in equationsinvolving photon energy. Note that the electron volt is a unit of energy andnot a unit of potential difference.

1 eV � 1.60 � 10�19 J

De Broglie Waves

Vocabulary De Broglie Wavelength: The effective wavelength of a moving particle.

Recall that photons of light exhibit both particle and wave properties.According to de Broglie, if a moving particle of matter has a high momentum,it exhibits wave properties and has a measurable wavelength.

hcl

(Planck’s constant)(speed of light)wavelength

cl

The de Broglie wavelength of any particle can be found from the equation

wavelength � or � �

Solved Examples

Example 1: Glenn is a DJ at his high school radio station WPAA, which broadcasts at afrequency of 91.7 MHz. When the station is on the air, how much energy doeseach emitted photon possess a) in joules? b) in electron volts?

Solution: The term MHz means megahertz or 106 Hz. Therefore, 91.7 MHzmeans 91.7 � 106 Hz.

a. Given: f � 91.7 � 106 Hz Unknown: E � ?h � 6.63 � 10�34 J �s Original equation: E � hf

Solve: E � hf (6.63 � 10�34 J �s)(91.7 � 106 Hz) � 6.08 � 10�26 J

b. This energy in joules can be converted into electron volts by dividing by1.6 � 10�19 J/eV.

E � � 3.80 � 10�7 eV

Example 2: Bart uses a helium-neon laser to align his telescope. The laser emits red lightwith a wavelength of 633 nm. How much energy, in electron volts, is given offby each photon of laser light?

Solution: First, convert nm to m. 633 nm � 6.33 � 10�7 m.

Given: h � 6.63 � 10�34 J �s Unknown: E � ?� � 6.33 � 10�7 m Original equation: E �c � 3.00 � 108 m/s

Solve: E � � � 3.14 � 10�19 J

This can be converted into electron volts by dividing.

E � � 1.96 eV

Example 3: Compare the de Broglie wavelengths for a proton and an electron, eachtraveling at 3.00 � 107 m/s.

3.14 � 10�19 J

1.60 � 10�19 J>eV

16.63 � 10�34 J # s2 13.00 � 108 m>s 26.33 � 10�7 m

hcl

hcl

6.08 � 10�26 J

1.60 � 10�19 J>eV

hmv

Planck’s constant(mass)(velocity)

234 Modern Physics

Given: mp � 1.67 � 10�27 kg Unknown: � � ?me � 9.11 � 10�31 kg Original equation: � �

v � 3.00 � 107 m/sh � 6.63 � 10�34 J �s

For the proton:

Solve: � � � � 1.32 � 10�14 m

For the electron:

Solve: � � � � 2.43 � 10�11 m

Therefore, the electron’s wavelength is 1800 times larger than the proton’s.

Practice Exercises

Exercise 1: The sun is a yellow star and emits most of its radiation in the yellow portionof the spectrum. If the sun’s radiation peaks at a frequency of 5.20 � 1014 Hz,how much energy is emitted by one photon of this visible yellow light?

Answer:

Exercise 2: After applying sunscreen, Cherie lies in the summer sun to get a tan. Theultraviolet light responsible for tanning has a wavelength over 310. nm, whilethe burning rays can range down to 280. nm. Which ultraviolet photons emitmore energy, those that tan or those that burn? How much more?

Answer:

Exercise 3: Gayle cooks a roast in her microwave oven. The klystron tube in the ovenemits photons whose energy is 1.20 � 10�3 eV. What are the wavelengths ofthese photons?

Answer:

Tanning rays: E � hc/� � (6.63 � 10�34 J �s)(3.00 � 108 m/s)/(310. � 10�9 m)

Burning rays: E � hc/� � (6.63 � 10�34 J �s)(3.00 � 108 m/s)/(280. � 10�9 m)

6.63 � 10�34 J # s19.11 � 10�31 kg 2 13.00 � 107 m>s 2

hmv

6.63 � 10�34 J # s11.67 � 10�27 kg 2 13.00 � 107 m>s 2

hmv

hmv

Modern Physics 235

Exercise 4: During the winter Olympic biathalon trials, Eric is shooting his rifle at atarget. What is the de Broglie wavelength of a 10.0-g bullet fired from the rifleat 500. m/s?

Answer:

Exercise 5: An electron microscope is observing detail on a virus down to 5.0 � 10�9 m.How fast must an electron in the microscope be moving to observe detail thissize? (Hint: Due to diffraction effects, an electron’s wavelength must be aboutthe same size or smaller than the object being observed.)

Answer:

18-2 The Photoelectric EffectBecause photons of light carry energy, they can cause electrons to be ejectedfrom certain metal surfaces just by being absorbed by the metal andtransferring their energy to the electrons. This process is known as thephotoelectric effect. However, certain conditions must be met in order forphotoelectrons to be ejected.

First, the incoming photon must have enough energy to cause the liberation ofan electron. The frequency that corresponds to this amount of energy is calledthe threshold frequency. At the threshold frequency, the photon has justenough energy to free the electron from the surface, and there is no excesskinetic energy given to the emitted electron.

The energy required to free the electron is called the work function. Anyexcess energy given to the electron becomes the kinetic energy that puts theelectron in motion. Therefore,

photon energy � kinetic energy � work function or hf � KE � W

The photoelectric effect is an interaction of one photon with one electron. Therelease of electrons from a surface is a function of the energy they receive.Therefore, you are more likely to witness the photoelectric effect by shiningdim blue light on a surface than bright red light, because the blue light has ahigher energy per photon.

236 Modern Physics

The wavelengths of colors in the visible light spectrum fall approximately inthe following ranges.

Violet light 400–440 nm Yellow light 530–590 nm

Blue light 440–480 nm Orange light 590–630 nm

Green light 480–530 nm Red light 630–700 nm

Radiation falling just below 400 nm is called ultraviolet radiation, while thatfalling just above 700 nm is called infrared radiation. Generally, thephotoelectric effect only occurs with ultraviolet and visible radiation.

Solved Examples

Example 4: When Doug walks through the entrance to the hardware store, a bell in theback of the store rings, triggered by a photocell whose work function is 2.70 eV. a) What is the threshold frequency of the light shining on thephotocell? b) What is the wavelength of the light?

a. Given: E � 2.70 eV Unknown: f � ?h � 6.63 � 10�34 J �s Original equation: E � hf

Solve: f � � � 6.51 � 1014 Hz

b. Given: c � 3.00 � 108 m/s Unknown: � � ?f � 6.03 � 1014 Hz Original equation: c � �f

Solve: � � � � 4.60 � 10�7 m � 460 nm This is blue light.

Example 5: What is the kinetic energy of photoelectrons emitted when ultraviolet light of200. nm shines on a photocell whose work function is 2.50 eV?

Given: h � 6.63 � 10�34 J �s Unknown: KE � ?c � 3.00 � 108 m/s

Original equation: � KE � W� � 2.00 � 10�7 mW � 2.50 eV

Solve: KE � � W

� � (2.50 eV)(1.6 � 10�19 J/eV)

� 5.95 � 10�19 J

16.63 � 10�34 J # s 2 13.00 � 108 m>s 22.00 � 10�7 m

hcl

hcl

3.00 � 108 m>s6.51 � 1014 Hz

cf

12.70 eV 2 11.60 � 10�19 J>eV 26.63 � 10�34 J # s

Eh

Modern Physics 237

Practice Exercises

Exercise 6: The work function for three surfaces are as follows: mercury � 4.50 eV,magnesium � 3.68 eV, and lithium � 2.30 eV. a) At what threshold frequencyare electrons liberated from each of these surfaces? b) What color lightcorresponds to these threshold frequencies?

Answer: a.

Answer: b.

Exercise 7: Shelley shines her red, helium–neon laser, whose wavelength is 633 nm, on aphotocell that has a work function of 2.38 eV. a) Will the photocell functionwith this wavelength of light? b) If so, what is the kinetic energy of thephotoelectrons released? If not, what wavelength corresponds to the thresholdfrequency?

Answer: a.

Answer: b.

Exercise 8: A classic physics demonstration involves placing a shiny zinc plate on anegatively charged electroscope and shining ultraviolet light on the plate. Ifthe work function of zinc is 4.31 eV and the wavelength of the light is 250 nm,with what kinetic energy are photoelectrons ejected from the zinc plate? b) What will happen to the leaves of the electroscope?

Answer: a.

Answer: b.

238 Modern Physics

18-3 Energy Level DiagramsEach atom has its own characteristic set of “fingerprints” or allowed energystates that its electrons can occupy. An energy level diagram is arepresentation of these allowed energy states. The electrons in an atom cannotoccupy any level between these allowed states, but instead “jump” from levelto level. This is analogous to a person trying to stand between, rather than on,the rungs of a ladder. It is impossible to do so!

Generally, electrons are found at the lowest energy level or ground state.However, when an electron absorbs a photon from its surroundings, itbecomes excited and jumps up to a higher energy level. Since the photon isremoved from the incident light, this produces an absorption spectrum.

When the electron returns to a lower energy level, it emits one or morephotons in the process, producing a bright line or emission spectrum. If theseemitted photons fall in the visible portion of the spectrum, the characteristicspectral lines of the material are seen.

The energy level diagram for hydrogen is shown. Because hydrogen is themost abundant gas in the universe, the hydrogen spectrum has been studiedvery closely and names have been given to the transitions between energy levels.

Lyman series: Electrons jump to or from the n � 1 level. The electromagnetic radiationemitted or absorbed is characterized as ultraviolet.

Balmer series: Electrons jump to or from then � 2 level. The electromagnetic radiationemitted or absorbed is characterized asvisible light.

Paschen series: Electrons jump to or from the n � 3 level.

Brackett series: Electrons jump to or from the n � 4 level.

Pfund series: Electrons jump to or from the n � 5 level. The electromagnetic radiationemitted or absorbed from these three seriesis characterized as infrared.

Modern Physics 239

Solved Examples

Example 6: a. What wavelengths of light are emitted by an electron jumping from n � 2to n � 1, and from n � 4 to n � 3? b) To what portion of the electromagneticspectrum do these wavelengths correspond?

From n � 2 to n � 1

Given: E2 � 10.2 eV Unknown: � � ?E1 � 0 eV Original equation: � E2 � E1

h � 6.63 � 10�34 J �sc � 3.00 � 108 m/s

Solve: � � � � 1.22 � 10�7m

From n � 4 to n � 3

Given: E4 � 12.75 eV Unknown: � � ?E3 � 12.09 eV Original equation: � E4 � E3

h � 6.63 � 10�34 J �sc � 3.00 � 108 m/s

Solve: � � � � 1.88 � 10�6 m

b. 122 nm is ultraviolet and 1880 nm is infrared.

Practice Exercises

Exercise 9: Use the energy level diagram for hydrogen on page 239 to determine theshortest wavelength in the Paschen series of hydrogen.

Answer:

16.63 � 10�34 J # s 2 13.00 � 108 m>s 2112.75 eV � 12.09 eV 2 11.60 � 10�19 J>eV 2

hcE4 � E3

hcl

16.63 � 10�34 J # s 2 13.00 � 108 m>s 2110.2 eV � 0 eV 2 11.60 � 10�19 J>eV

2hcE2 � E1

hcl

240 Modern Physics

Exercise 10: The sun’s spectrum is made up of many absorption lines called Fraunhofer lines.How many electron volts of energy are absorbed in order to produce the H line whose wavelength is 657.7 nm?

Answer:

Exercise 11: A stellar spectrum shows three absorption lines of hydrogen produced aselectrons move from the n � 2 state to higher energy levels (n � 3, n � 4,n � 5). What are the wavelengths and colors of the three lines missing fromthe continuous spectrum?

Answer:

Exercise 12: On June 24, 1999, NASA launched FUSE (the Far Ultraviolet SpectroscopicExplorer) to explore the universe using high-resolution spectroscopy in the farultraviolet spectral region. If FUSE records radiation of wavelength 102.8 nm,between what two energy levels must the electron jump in the hydrogen atomto produce this line?

Answer:

18-4 RadioactivityMany atoms that contain large numbers of neutrons are unstable, orradioactive. This means that in a given amount of time, the atoms of thesubstance will decay or change into different atoms with the emission of �, �,or rays from the nucleus.

Vocabulary Activity: The rate at which a radioactive sample decays.

Modern Physics 241

If a radioactive sample containing N atoms is allowed to decay for an elapsedtime, �t, there will be a change in the number of atoms, �N, which dependsupon the decay constant, �, for that particular material. Note: This is not thesame � used to represent wavelength. The decay constant is the probabilityper unit time that a nucleus will decay. The term �N/�t is called the activity.

� �(decay constant)(original number of atoms)

or � ��N

The SI unit for activity is the becquerel (Bq), which equals one decay persecond.

The number of radioactive atoms, N, remaining after a time, t, can be found ifyou know the number of atoms in the original sample, No, and the decayconstant of the material, �.

N � No e��t

where e is the base of the natural logarithm and is approximately equal to 2.72.

Another way of examining radioactivity is by looking at the half-life of asample.

Vocabulary Half-life: The time it takes for half of a radioactive sample to decay.

Half-life � or T1/2 �

Solved Examples

Example 7: Cobalt-60, used in radiation therapy for cancer patients, has a half-life of 5.26 y. A sample of cobalt-60 containing 5.00 � 1012 radioactive atoms sits in alead case in the medical stockroom of St. Mary’s Hospital for 10.0 years. Howmany cobalt-60 atoms remain after this amount of time?

Solution: First, find the decay constant for cobalt-60.

0.693l

0.693decay constant

¢N¢t

change in number of atomselapsed time

242 Modern Physics

� � � � 0.132 y�1

Given: � � 0.132 y�1 Unknown: N � ?No � 5.00 � 1012 atoms Original equation: N � Noe��t

e � 2.72t � 10.0 y

Solve: N � Noe��t � (5.00 � 1012 atoms)(2.72)�(0.131 y�1)(10.0 y)

� 1.33 � 1012 atoms

Example 8: Radioactive gold-198 is used as a tracer to test liver functions in low-levelliver scans. Dr. Rogers uses gold-198 in a liver scan on Otis, who has beenexhibiting signs of jaundice. A solution containing 3.00 � 109 gold-198 atomsis injected into his liver and observed after 72.0 h. What is the activity of thegold-198 after this amount of time? (Half-life of gold-198 � 2.70 d)

Solution: First, convert days into hours. 2.70 d � 64.8 h

Next, find the decay constant of gold-198.

� � � � 0.0107 h�1

Then, find the number of gold-198 atoms remaining after 72.0 h.

Given: No � 3.00 � 109 atoms Unknown: N � ?e � 2.72 Original equation: N � Noe��t

� � 0.0107 h�1

t � 72.0 h

Solve: N � Noe��t � (3.00 � 109 atoms)(2.72)�(0.0107 h�1)(72.0 h) � 1.39 � 109 atoms

Finally, calculate the activity after converting h into s. 64.8 h � 233 280 s

� � � 2.97 � 10�6 s�1

Given: � � 2.97 � 10�6 s�1 Unknown: �N/�t � ?N � 1.39 � 109 atoms Original equation: �N/�t � ��N

Solve: �N/�t � ��N � �(2.97 � 10�6 s�1)(1.39 � 109 atoms) � �4130 Bq

0.693233 280 s

0.69364.8 s

0.693T1>2

0.6935.26 y

0.693T1>2

Modern Physics 243

Practice Exercises

Exercise 13: Spent fuel rods contain strontium-90 whose half-life is 28.1 y. Josh works at anuclear reactor and must safely store the spent rods. If a spent fuel rodcontains 1.00 � 1027 atoms of strontium-90 when stored in a sealed container,how many strontium-90 atoms will remain if the container is excavated byarcheologists 1000. y later?

Answer:

Exercise 14: The synthetically manufactured radiopharmaceutical technicium-99 is used to produce a scan of Dale’s brain after he suffers a concussion. The half-life oftechnicium-99 is 6.02 h. What percent of technicium-99 remains in Dale’s body24 h after the scan?

Answer:

Exercise 15: In the movie The Planet of the Apes, the forbidden zone was an areapresumably contaminated by the radioactive plutonium fallout from thedetonation of nuclear weapons. If Zera finds a rock in the forbidden zone thatis tainted with plutonium-239 whose activity is 100. Bq, how many atoms ofplutonium does the rock contain when it is discovered? (Half-life ofplutonium-239 is 24 900. y)

Answer:

Exercise 16: In Exercise 15, if the explosion occurred 500. y prior to Zera’s discovery, howmany plutonium-239 atoms did the rock originally contain?

Answer:

244 Modern Physics


A-1: Gamma rays emitted during the explosion of a nuclear bomb have an energy of 1.2 � 106 eV per photon. What is the frequency of this gamma ray emission?

A-2: An X-ray technician always steps out of the room when the X-ray machine ison. How much energy is carried by each photon of X-ray radiation, if thewavelength of this radiation is 0.0800 nm?

A-3: Mitch is undergoing eye surgery to repair a detached retina. His doctor uses agreen laser whose wavelength is 514 nm. How much energy is produced byeach laser photon?

A-4: Roy is making holograms with his helium-neon laser. In a helium-neon laser,excited helium atoms collide with neon atoms, raising the neon to an excitedstate where its energy is 20.66 eV. Stimulated emission then causes electrons inthe neon to drop to a lower energy level where E � 18.7 eV. What is thewavelength and color of the light given off by a helium-neon laser?

A-5: At Bell Labs in 1926, Davisson and Germer aimed a beam of electrons at anickel crystal whose atomic spacing was 0.215 nm. If the electrons had aspeed of 4.4 � 106 m/s, calculate the de Broglie wavelength of the electrons todetermine whether they would be able to pass through the crystal structure orwould reflect back.

A-6: To determine the size of an oxygen nucleus, protons with kinetic energy of0.100 GeV (1.00 � 108 eV) are shot at oxygen atoms. a) How fast are theprotons moving? b) What is the de Broglie wavelength of the proton?

A-7: Three surfaces, sodium, iron, and gold, have respective work functions of 2.46 eV, 3.90 eV, and 4.82 eV. If light, whose wavelength is 300. nm, shines oneach of these materials, which ones will show the photoelectric effect, andwhat will be the kinetic energy of any photoelectrons emitted?

A-8: Use the energy-level diagram for mercury to eVdetermine how much energy is needed to ionize a mercury atom in the n � 4 level.

A-9: A mercury atom absorbs a photon of wavelength 161 nm. What energy level does itjump to?

A-10: Hal looks at a mercury vapor street lamp through a diffraction grating and measuresthe wavelength of a spectral line to be 577 nm.Between what two energy levels must theelectron jump to produce this line?

Modern Physics 245

Energy levels for Mercury

A-11: Gloria is testing her basement for radon with a kit she bought at thedrugstore. The half-life of radon is 3.83 days and Gloria is informed that5.00 � 106 radon atoms were present in her basement at the time of testing.Gloria hires a mason to seal off her basement and she runs the test again 30.0 days later. How many radon atoms will now be found in the basement?

A-12: While checking the radioactive tritium levels in the missiles at a Titan missilesite, Hugh discovers that it has been 15.0 years since one of the missiles waslast inspected. What percent of the radioactive tritium has been depleted?(Half-life of tritium � 12.4 y)

A-13: Nina has a watch whose hands glow in the dark due to a special paintcontaining radium-226 whose half-life is 1.60 � 103 y. When Nina takes thewatch in for a cleaning after 20.0 y, the radium in the hands is found to havean activity of 1.12 � 1014 Bq. How many radium-226 atoms does the watchcontain at this time?

A-14: In A-13, how many radium-226 atoms were originally in the watch when itwas first purchased 20. years ago?


B-1: In Exercise 5, what minimum accelerating voltage of the electron microscopewill produce an electron with this de Broglie wavelength?

B-2: Carbon-14 is commonly used to determine the age of organic material.Darlene is on an archeological dig in Mexico and discovers among some ruins what she thinks is an ancient Mayan bone. a) If the bone shows activityof 2.59 � 106 Bq, while the same mass of new human bone shows an averageactivity of 3.11 � 106 Bq, how old are the excavated bones? (Hint: Half-life of C-14 � 5730 y.) b) Why is carbon-14 a good substance to use for radioactive dating?

B-3: Two containers of radioactive iodine sit on a shelf in Doctor Bailin’s supplycloset, but the print on the labels has faded and is difficult to read. Dr. Bailinneeds some iodine for a thyroid scan but she must only use iodine-131 whosehalf-life is 8.27 h and not iodione-59 whose half life is 44.6 d. She tests asample and finds its activity to be 5.00 � 105 Bq. What should the activity be24.0 h later if Dr. Bailin is testing the iodine-131?

246 Modern Physics

Appendix A:WORKING WITH NUMBERS

Significant Figures

Addition and Subtraction: When adding or subtracting numbers, youranswer cannot have more significant figures after the decimal than thesmallest number of significant figures after the decimal in any of the numbersused to obtain the answer.

For example: 25.678 � 3.45 � 67.2 � 96.3 Only 3 significant figures

Multiplication and Division: When multiplying or dividing numbers, youranswer cannot have more total significant figures than the smallest totalnumber of significant figures in any of the numbers used to obtain the answer.

For example: (26.4 N)(1.2 m) � 32 N �m Only two significant figures

These rules are fairly easy to follow until you begin introducing zeros intoyour equations. Below are some examples using zeros.

700 has only one significant figure (the 7).700.0 has 4 significant figures (all 4 numbers).0.0700 has 3 significant figures (the 7 and the two zeros to the right of the 7).0.007 has only 1 significant figure (the number 7).7.007 has 4 significant figures (all 4 numbers).

Scientific notation may come in handy when working with significant figures.The number 7000, which has only 1 significant figure, can be written as7.00 � 103 in order to be written with three significant figures.

You may also make zeros significant by placing a decimal point at the end.For example, 700 has only one significant figure while 700. has three.

Remember, these rules only apply to measured quantities. Quantities that canbe counted rather than measured, such as people, coins, etc., are presumed tobe an exact number and may be followed by as many zeros after the decimalas needed.

Unit Conversions

Before you solve an exercise, it is important that all units on the ends of thenumbers you are using be in the same system. In this book, most quantitieshave been converted into the SI System (Système International), which is thestandard system of units in physics.

Appendix 247

248 Appendix

Example: Convert 5 years into seconds.

(5 ) � � � �

Notice that anything you are trying to eliminate in the numerator must bewritten in the denominator and vice versa. The units appearing in both thenumerator and denominator cancel each other out, as shown by the slashmarks through them.

Multiplying the numerators gives: 5 � 365 � 24 � 60 � 60 � 157 680 000

Multiplying the denominators gives: 1 � 1 � 1 � 1 � 1

Final answer: � 157 680 000 s Note: This number is not written

with significant figures.

Some Simple Trigonometry Relationships

The rules of trigonometry are developed with the use of right triangles as shown in thelabeled diagram. Using this diagram, you canconstruct trigonometric equations in thefollowing way.

sin � � cos � � tan � �

Remember, you can only use the above relationships with right triangles. Thehypotenuse of a right triangle is always the longest side.

Some Common Prefixes

Mega (M) � 1 � 106 centi (c) � 1 � 10�2 micro (�) � 1 � 10�6

kilo (k) � 1 � 103 milli (m) � 1 � 10�3 nano (n) � 1 � 10�9

opposite

adjacent

adjacent

hypotenuse

opposite

hypotenuse

157 680 000 s1

160 s 211 min 2

160 min 211 h 2

124 h 211 d 2

1365 d 211 y 2y

Chapter 11. 884 m3. a) 10.7 s5. �9.00 m/s2

7. a) 2.2 m/s9. a) 28.3 m/s

11. 0.36 s13. 25.9 m

A1. 3.78 � 108 mA3. a) 16.61 m/s

b) 37.16 mi/hA5. Tortoise wins by 47 sA7. 5 sA9. �19 500 m

A13. a) 1.11 sb) The same

A15. 3.9 m/s2

Chapter 21. 145 km south3. 50. N forward5. 175 m/s northeast7. 6360 km9. Horizontal: 752 N

Vertical: 274 N11. 40.0 m/s13. 37° to horizontal15. 465 m

A1. a) 1450 km southb) 1650 km

A3. 4.1 m/s76° east of north

A5. 14.1 m45° north of west

A7. Horizontal: 50. NVertical: 42 N

A9. 4.2 mA11. 13.4 m/sA13. 1.2 m (does not clear)

Chapter 33. 4500 N5. a) �34 000 N7. 102 N9. a) 78 N

11. 0.2013. 14.5 N15. 10 600 N17. 1860 N19. Culp: 1.5 � 107 N/m2

Vance: 1.0 � 106 N/m2

21. a) .0079 m2

b) 0.050 mA1. 50. NA3. �1400 NA5. 100 NA7. 0.20A9. 17.7 N

A11. 625 NA13. 0.080 NA15. a) 37 500 N

b) 75 000 N/m2

Chapter 41. 4.91 � 109 kg �m/s3. a) 3750 N

b) 1.5 � 106 Nc) 400 times as great

5. 11.3 m/s7. a) 9.0 m/s9. 10. m/s

11. 215 m/s13. 9.42 m/s

A1. 1470 kg �m/sA3. 6080 NA5. 50 000 NA7. 1.04 m/sA9. �0.22 m/s

A11. 17 004 m/s

Chapter 51. a) 1430 J3. 4 N5. 600 J7. 17.4 m/s9. 24.5 m/s

11. 7.713. 280 N15. a) 6.0

b) 4.4c) 73%

A1. 18 800 JA3. a) 1 610 000 J

b) 53 700 Wc) 1 040 000 J

A5. 12 JA7. a) 1.6 JA9. 15 200 m

A11. 38A13. 3

Chapter 61. 0.034 s3. a) 1.6 m/s5. a) 0.63 m/s7. a) Jessica: 1.3 m/s

Julie 0.94 m/sb) 0.15 m/s2

9. 2.5 m/s11. 1.50 � 10�15 N13. 0.12 N �m15. a) 17 N up17. Anita: 384 N up

Orin: 246 N up19. 9.74 � 1037 kg �m2

21. 5.4 � 10�8 kg �m2

23. 5.6 m/sA1. 0.6 sA3. 3.8 mA5. a) 2.2 m/s

b) 35 NA7. a) 5.9 � 10�3 m/s2

b) 3.5 � 1022 N toward sunc) 3.5 � 1022 N toward

EarthA9. Outstretched: 12 N �m

Bent: 5.6 N �mA11. 1.5 � 10�4 kg �m2A13. 18 m/s

Chapter 71. Twice as large.3. a) 9.78 � 10�8 N

b) 1.63 � 10�8 N5. 4.2 � 1021 m7. 1.67 � 10�9 m/s2

9. a) 9.9 � 1030 kgb) 5.0 times

11. a) 618 000 m/s13. a) 2.58 � 10�4 m/s

A1. 1/8100A3. 4.1 � 10�47 NA5. a) 1.66 � 10�3 NA7. a) 4.9 � 1023 kgA9. 1.33 � 10�9 m/s2

A11. a) 1340 Nb) 3560 m/s

Chapter 81. 2 m/s3. 28 h5. Albert 33 y

Henry 39.6 y7. 13.0 m9. a) 94 450 km

11. 1.0 � 10�3 kgA1. 15 m/s same directionA3. 8.3 yA5. 1390 mA7. 1068 J

Chapter 91. a) 710. kg/m3

3. 1.76 � 10�3 kg5. 1.3 � 108 N/m2

7. 5.3 � 10�3 m2

9. a) 3.0 � 10�6 m11. 4.2 � 106 Pa13. 1330 kg/m3

15. 1.61 N17. 0.50 m19. Decreases by 314 m3

21. 0.015 m3

A1. 880 kg/m3

Appendix 249

Appendix B:SELECTED ANSWERS

A3. Silver: 10 500 kg/m3

Earth: 5540 kg/m3

A5. 1.5 � 10�3 mA7. b) 21 mA9. a) 9.0 � 103 N

b) 10. NA11. a) 3.03 times bigger

Chapter 101. 122°F3. a) �148°C

b) �234°F5. 437°C7. b) 3.96 � 10�4 m3

9. 347 000 J11. 31.8°C13. 9630 J15. 3.9 � 106 J

A1. Hottest: 462°CColdest: �218°C

A3. 2.0 � 10�3 mA5. 3.1 � 10�4 m2

A7. 1990 cm3

A9. 79.7°CA11. 0.019 kgA13. 1300 J

Chapter 111. 0.67 s3. a) 200 N/m5. a) 0.63 s7. 2 s9. a) 3.1 s

11. 0.65 mA1. 0.0023 sA3. 20 N/mA5. a) 0.5441 sA7. 6.28 sA9. b) 0.0400 m

Chapter 121. 0.013 m3. 0.4 m/s5. 188 Hz7. b) �5.26 m/s9. 813 Hz

11. 0.300 m13. 394.0 Hz

A1. 0.0085 mA3. a) 0.688 mA5. 40 mA7. a) Toward: 501.5 Hz

Away: 498.5 Hzb) 3.0 Hz

A9. 15.0 m/sA11. 20.4 m/sA15. 628 m/s

Chapter 131. 19 700 s3. 3.80 � 10�7 m5. 40°7. a) �36 cm9. 28.9°

11. 1.39A1. 260 s

A3. 20°A5. c) �12.0 cmA7. a) A9. b) alcohol: 2.21 � 108 m/s

water: 2.26 � 108 m/sA11. 20.7°A13. a) 42.5°

Chapter 141. 11.1 cm3. a) 0.0508 m5. a) 20 times

b) 30 cm7. 8 times9. b) 2.7 diopters

11. 0.40 m13. a) 8.8 � 10�4 m15. 2.4 � 10�6 m

A1. 12.0 cmA3. 2.4 cmA5. 24 cmA7. 0.17 mA9. a) �4.0 diopters

b) 0.29 mA11. �0.17 dioptersA13. 7130 m

Chapter 151. 1.3 � 10�3 N3. 7.0 � 10�13 C5. 10. m7. 1.9 � 1013 N/C9. 18 � 105 N/C to the right

11. 8800 V13. 4.0 � 10�3 m

A1. 1.5 � 10�11 NA3. 4.2 � 10�13 CA5. a) 0.043 mA7. 1.3 � 107 N/CA9. a) 0.14 m

A11. 450 000 JA13. 600 000 V

Chapter 161. 10 800 s3. 27.5 �5. 1.52 V7. 2.0 � 104 V9. 0.8 A

11. a) 240 Vc) 18 A

13. a) 1.5 Ab) 80 �

15. $8.1017. a) 3 �

b) 4 A19. Series: 10. V

Parallel: 120 V21. a) 5 �

b) 85 �A1. a) 15 600 sA3. 100. �A5. 5 � 10 �9 AA7. 14.17 AA9. $1.2 � 106

A11. 0.25 AA13. a) 3.0 �

b) 12 A

Chapter 171. 4.2 � 10�14 N3. a) 6.0 � 10�3 T5. a) Zero

b) 1.4 � 10�5 Wb7. 8.4 � 10�8 V9. a) 12 V

A1. 1.1 � 10�10 NA3. 8.6 � 10�14 NA5. a) 1.3 � 10�3 N

b) zeroA7. a) 1.1 � 10�5 VA9. a) 4800 V

Chapter 181. 2.15 eV3. 1040 nm5. 150 000 m/s7. a) 1.96 eV

b) 5.22 � 10�7 m9. 823 nm

11. a) 657 nm Redb) 488 nm Greenish Bluec) 445 nm Violet

13. 1.85 � 1016 atoms15. 1.13 � 1014 atoms

A1. 2.9 � 1020 HzA3. 2.42 eVA5. 1.7 � 10�19 mA7. Sodium: 1.68 eV Yes

Iron: 0.244 eV YesGold: �0.676 eV No

A9. 7.72 eVA11. 2.18 � 104 atomsA13. 8.18 � 1024 atoms

250 Appendix

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